\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps,pgfshade} \usepackage{tikz} \usetikzlibrary{arrows,automata,backgrounds} \usetikzlibrary{shapes} \usepackage{pgfbaseimage} \tikzstyle{block}=[draw opacity=0.7,line width=1.4cm] \usetikzlibrary{calc,3d} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf Space-Efficient Generation of \\ \vskip .1in Nonisomorphic Maps and Hypermaps } \vskip 1cm \large Timothy R. Walsh \\ Department of Computer Science\\ UQAM \\ Box 8888, Station A \\ Montr\'eal, Quebec, H3C 3P8 \\ Canada \\ \href{mailto:walsh.timothy@uqam.ca}{\tt walsh.timothy@uqam.ca}\\ \end{center} \vskip .2 in \begin{abstract} In 1979, while working as a senior researcher in the Computing Centre of the USSR Academy of Sciences in Moscow, I used Lehman's code for rooted maps of any orientable genus to generate these maps. By imposing an order on the code-words and keeping only those that are maximal over all the words that code the same map with each semi-edge chosen as the root, I generated these maps up to orientation-preserving isomorphism, and by comparing each of them with the code-words for the map obtained by reversing the orientation, I generated these maps up to a generalized isomorphism that could be orientation-preserving or orientation-reversing. The limitations on the speed of the computer I was using and the time allowed for a run restricted me to generating these maps with up to only six edges. In 2011, by optimizing the algorithms and using a more powerful computer and more CPU time I was able to generate these maps with up to eleven edges. An average-case time-complexity analysis of the generation algorithms is included in this article. And now, by using a genus-preserving bijection between hypermaps and bicoloured bipartite maps that I discovered in 1975 and the condition on the word coding a rooted map for the map to be bipartite, I generated hypermaps, both rooted and unrooted, with up to twelve darts (edge-vertex incidence pairs). \end{abstract} \section{Introduction} A \emph{map} is defined topologically as a 2-cell embedding~\cite{Arq87a} of a connected graph, loops and multiple edges allowed, in a 2-dimensional surface. The \emph{faces} of a map are the connected components of the complement of the graph in the surface. In this article the surface is assumed to be without boundary and orientable, with an orientation already attributed to it (counter-clockwise, say), so that it is completely described by a non-negative integer $g$, its \emph{genus}. For short, a map on a surface of genus $g$ will be called a \emph{map of genus $g$}. A \emph{planar map} is a map of genus 0 (a map on a sphere). If a map on a surface of genus $g$ has $v$ vertices, $e$ edges and $f$ faces, then by the Euler-Poincar\'e formula~\cite[Chap. 9]{Cox73} \begin{equation} f - e + v = 2(1-g). \label{eq:1} \end{equation} Two maps are \emph{equivalent} if there is an orientation-preserving homeomorphism between their embedding surfaces that takes the vertices, edges and faces of one map into the vertices, edges and faces of the other. A \emph{dart} or \emph{semi-edge} of a map or graph is half an edge. A loop is assumed to be incident twice to the same vertex, so that every edge, whether or not it is a loop, contains two darts. The face incident to a dart $d$ is the face incident to the edge containing $d$ and on the right of an observer on $d$ facing away from the vertex incident to $d$. A \emph{rooted map} is a map with a distinguished dart, its \emph{root}. Two rooted maps are equivalent if there is an orientation-preserving homeomorphism between their embedding surfaces that takes the vertices, edges, faces and the root of one map into the vertices, edges, faces and the root of the other. A \emph{combinatorial map} is a connected graph with a cyclic order imposed on the darts incident to each vertex, representing the order in which the darts of a (topological) map are encountered during a rotation around the vertex according to the orientation of the embedding surface. Given a dart $d$, we denote by $-d$ the other half of the edge containing $d$ and by $P(d)$ the next dart after $d$ according to the cyclic order of the darts around the vertex incident to $d$. The darts incident to a face are encountered by successive application of the permutation $P-$ (${-}~$ followed by $P$). In this way the faces of a combinatorial map can be counted, so that its genus can be calculated from~(\ref{eq:1}). Two combinatorial maps are equivalent if they are related by a \emph{map isomorphism} -- a graph isomorphism that preserves this cyclic order -- with an analogous definition for the equivalence of two rooted combinatorial maps. An \emph{automorphism} of a combinatorial map is a map isomorphism from a map onto itself. Following~\cite{Wormald81a} and~\cite{Wormald81b}, we define a \emph{sensed map} to be an equivalence class of maps and an \emph{unsensed map} to be an equivalence class of maps under a homeomorphism that could be orientation-preserving or orientation-reversing. It was shown in~\cite{JoSi} that each equivalence class of topological maps is uniquely defined by an equivalence class of combinatorial maps; so from now on a rooted map means a rooted combinatorial map, a sensed map means an isomorphism class of combinatorial maps and an unsensed map means an equivalence class of maps under both isomorphism and reversal of the cyclic order imposed on the darts incident to each vertex. Map enumeration began in earnest with the work of Tutte, who used it in an attempt to solve the famous four-colour problem. Lehman used it in his study of the molecular structure of polymers. In addition, map enumeration has applications in classical and algebraic combinatorics~\cite{GJ83}, theoretical physics and integrable hierarchies~\cite{LZ04}. There are many research papers on the enumeration of maps with various properties; we list here some of the papers in which maps (rooted, sensed and unsensed) have been enumerated by genus and either number of edges alone or number of edges and vertices (the latter is equivalent by (\ref{eq:1}) to enumerating by number of faces and vertices). Rooted planar maps were counted by Tutte, first by number of edges alone (as a closed-form formula)~\cite{Tu1} and then by number of faces and vertices (as a generating function)~\cite{Tu3}. I found an algorithm for counting rooted maps by genus, number of edges and number of vertices~\cite{Walsh71,W1} and a polynomial algorithm for counting rooted \emph{toroidal} maps (maps of genus 1), both by number of edges and by number of vertices and faces~\cite{Walsh83}. Using an improved version of the method of~\cite{Walsh71}, presented by Bender and Canfield~\cite{BC86}, Arqu\`es found a closed-form formula for counting rooted toroidal maps, both by number of edges and by number of vertices and faces~\cite{Arq87a}. Bender and Canfield found a closed-form formula for counting rooted maps of genus 2 and 3 by number of edges~\cite{BC91}. Giorgetti, a student of Arqu\`es, generalized the results of~\cite{Arq87a} and~\cite{BC91} to obtain a general form for the generating function counting rooted maps of any genus by number of vertices and faces and counted the maps of genus 2 and 3~\cite{gio98a}. I then collaborated with Giorgetti to extend this enumeration up to genus 6~\cite{WG14} and later up to genus 10~\cite{WGMunrooted}. Liskovets found a closed-form formula for the number of sensed planar maps by number of edges~\cite{Liskovets81}. Mednykh and Nedela generalized Liskovets' method and thus counted sensed maps of genus 1, 2 and 3 by number of edges~\cite{MN06} and then Giorgetti and Mednykh counted sensed maps of genus 4 by number of edges~\cite{MG11}. Then I collaborated with Giorgetti and Mednykh to count sensed maps of genus up to 10 by number of vertices and faces and up to genus 11 by number of edges~\cite{WGMunrooted,Walsh13}. Using a more efficient method for counting rooted maps discovered by Carrell and Chapuy~\cite{CC14-v3}, Giorgetti and I enumerated rooted and sensed maps of genus up to 50 with up to 100 edges in~\cite{GW14}, which includes tables of numbers of sensed maps of genus up to 19. And Wormald found an algorithm for counting planar maps, both sensed and unsensed, by number of edges and by number of vertices and faces~\cite{Wormald81a,Wormald81b}. The methods used to obtain all of the above results are computationally more efficient than exhaustive generation. But, as far as I know, exhaustive generation is the only method yet known to enumerate unsensed non-planar maps, and even for maps that have been enumerated by other methods, exhaustive generation serves to verify the numbers obtained by these methods. The method I used in~\cite{Walsh83b} to generate isomorphism classes of maps without having to store all the previously generated rooted maps to see whether each new map is isomorphic to one of the old ones is essentially the one used by Read~\cite{Read78} to generate the isomorphism classes of 9‑-vertex graphs. He generated the adjacency matrix of each of the labelled 9-vertex graphs and then eliminated all those that are not lexicographically largest among those matrices representing the same graph but with a different labelling. Since a rooted map has only the trivial automorphism~\cite{Tu1}, I generated all the rooted maps, or rather Lehman's code for rooted maps, with $e$ edges and $v$ vertices, eliminated all those whose code-word is not lexicographically largest among those coding the same map but with a different root, and sorted the rest by genus to generate sensed maps; to generate unsensed maps, I eliminated each sensed map whose code-word could be made lexicographically larger by reversing the cyclic order of the darts at each vertex and choosing one of the darts as the root. To be sure, more sophisticated methods of generating isomorphism classes of combinatorial objects have since been discovered~\cite{GLM97}, and for objects with many distinct labellings these methods are probably much faster. However, a map with $e$ edges has at most $2e$ distinct rootings; so the admittedly old-fashioned method I used seems to be quite adequate. More recently Jackson and Visentin published an atlas of maps~\cite{JV01}. A (combinatorial) \emph{hypermap} is a generalization of a map in which an edge is allowed to have any positive number of darts instead of exactly two and the darts are cyclically ordered around the edges as well as the vertices. In 1975 I published a genus-preserving bijection between hypermaps with $d$ darts, $e$ edges, $v$ vertices and $f$ faces and bicoloured bipartite maps with $d$ edges, $e$ black vertices, $v$ white vertices and $f$ faces, each containing twice as many darts as the corresponding face of the hypermap~\cite{Walsh75}. This bijection was used by Arqu\`es to count rooted planar~\cite{Arq86a} and toroidal~\cite{Arq87} hypermaps by number of vertices, edges and faces; Chauve~\cite{Chauve03} independently counted rooted bicoloured bipartite planar maps with the corresponding parameters. And now I discovered a condition on the Lehman word that codes a rooted map for the map to be bipartite, which I used to generate rooted, sensed and unsensed hypermaps with up to 12 darts. The words with which Lehman coded rooted maps are described in Section~\ref{sec2}, the procedure I used to generate these words is described and analyzed in Section~\ref{sec3}, a discussion of the generation of hypermaps appears in Section~\ref{sec4} and the results of the computation, including timings, are described in Section~\ref{sec5}. A table of numbers of unsensed maps with up to 11 edges, sorted by genus and number of vertices, appears in Appendix~\ref{appA}; the analogous tables for rooted maps and sensed maps appear in other sources, which are cited in Section~\ref{sec5}. Appendix~\ref{appB} contains a table of numbers of rooted, sensed and unsensed hypermaps. \section{Lehman's code for rooted maps} \label{sec2} In the 1960s Lehman, who was then my Ph.~D. supervisor, generalized the code for a rooted plane tree as a balanced parenthesis system to a code for a rooted planar map with a given spanning tree as a (balanced) parenthesis system (coding the rooted plane tree obtained by deleting the edges not in the spanning tree) merged with a bracket system (coding the rooted one-vertex map obtained by contracting the edges of the spanning tree). The number of pairs of parentheses is the number of edges of the spanning tree and the number of pairs of brackets is the number of edges not in the spanning tree. To code a rooted planar map without a spanning tree, he used Tamari's maze-running algorithm~\cite{Tarry95}, which is essentially depth-first search~\cite{Tarjan72} with the darts incident to each vertex encountered in their cyclic order, to construct a canonical spanning tree, and he proved that a spanning tree is canonical if and only if the code word for the rooted map with this spanning tree does not contain the forbidden sub-word [(]), where the right bracket is the \emph{mate} of (that is, closes) the left bracket, the right parenthesis is the mate of the left parenthesis and the four symbols are not necessarily contiguous. To code a rooted map of any orientable genus, he replaced the bracket system by an \emph{integer system on $m$ pairs}: a word consisting of two copies of each of the integers 1, 2, \ldots, $m$, where $m$ is the number of edges in the rooted one-vertex map coded by the integer system and the first occurrences of the integers are in increasing order. The forbidden sub-word is now $i(i)$, where the right parenthesis is the mate the left one. Each letter in a word coding a rooted map represents a dart, with the first letter representing the root. If a dart $d$ is (represented by) a parenthesis or a bracket, then $-d$ is its mate; if $d$ is an integer $i$, then $-d$ is the other occurrence of $i$. If $d$ is a bracket or an integer, then $P(d)$ is the next letter in the word (with wraparound); if $d$ is a parenthesis, then $P(d)$ is the letter that follows the mate of $d$ (with wraparound). The darts of the face containing $d$ can be found from the code-word by successive application of the permutation $P-$ to the letters representing the darts. For example, in the code word 123123, the face containing the first 1 also contains the second 2 and the first 3 (the next dart would be the first 1) and the face containing the first 2 also contains the second 3 and the second 1; since all the letters belong to one of these two faces, there are only 2 faces and so by (\ref{eq:1}) the one-vertex map coded by this word is of genus 1. Since contracting an edge does not change the genus of a map, the genus of a rooted map can be calculated from the integer sub-system of its code-word. A more detailed description of Professor Lehman's code, including his method of coding a rooted map, can be found in my Ph.~D. thesis~\cite{Walsh71} and in~\cite{Walsh83b}, where I described the use I made of his code to generate isomorphism classes of maps. \section{Generating maps} \label{sec3} To generate the rooted plane trees with $e$ edges, I generate the parenthesis systems with $e$ pairs of parentheses in lexicographical order, with a left parenthesis represented by 0 and a right parenthesis represented by -1. To generate the rooted planar one-vertex maps with $e$ edges, I generate the bracket systems with $e$ pairs of brackets, also in lexicographical order, with a left bracket represented by 2 and a right bracket represented by 1. To generate the not-necessarily-planar rooted one-vertex maps with $e$ edges, I generate the integer systems on $e$ pairs; in~\cite{Walsh83b} I made the second occurrence of each integer move from its leftmost position (immediately to the right of the first occurrence of the same integer) to its rightmost position (the rightmost letter in the word) with $e$ moving the fastest, whereas now I use a Gray code in which they move alternately to the right and to the left. Each new system is generated in $O(e)$ time in the worst case and $O(1)$ time in the average case. To generate the rooted maps with $e$ edges and $v$ vertices, I first generate the bracket systems or the integer systems on e-v+1 pairs, and in the latter case I calculate the genus by counting the faces (in $O(e)$ time) and substituting into (\ref{eq:1}) as described above. For each bracket system or integer system I generate all the parenthesis systems on $v-1$ pairs. For each pair of words I merge them in all possible ways that avoid the forbidden sub-word, moving each parenthesis from left to right, with a right parenthesis starting adjacent to its mate and stopping when it hits an integer or bracket whose mate is to the left of the parenthesis' mate or when it passes the last integer or bracket. The procedure for passing from one merged word to the next is described in more detail in~\cite{Walsh83b}. This procedure involves deleting a parenthesis when it reaches its rightmost position and then, when a parenthesis has been moved to the right, inserting all the deleted parentheses in their leftmost positions. Since in the worst case all the parentheses may get deleted and reinserted in passing from one word to the next, the algorithm runs in $O(e^2)$ worst-case time if the letters following a deleted parenthesis are pulled to the left as in~\cite{Walsh83b}. Now I replace each deleted parenthesis by a marker ($-$2). After a parenthesis has been moved to the right, some of the slots between successive undeleted parentheses (or to the left of the leftmost parentheses or to the right of the rightmost one) will contain both markers and either integers or brackets. In each such slot I move all the markers to the left side of the slot and all the integers or brackets to the right side and then replace all the markers by the deleted parentheses, so that the algorithm runs in $O(e)$ worst-case time. To generate the sensed maps with $e$ edges and $v$ vertices, I generate the rooted maps with $e$ edges and $v$ vertices, or rather, their Lehman code-words, and then I check each one for lexicographical maximality with respect to the code-words for all the rootings of the same map. To this end, I decode the code-word into a rooted map represented by two arrays VERT and NEXT, where the darts are the indices $1, 2, \ldots, 2e$, the $i$th edge encountered during the decoding procedure consists of the darts $i \leq e$ and $2e+1-i$, VERT[$i$] is the label assigned to the vertex containing the dart $i$, NEXT[$i$] is $P(i)$ and the root is dart 1. Then, I code this map rooted at each of the other darts and compare the new code-word with the original one. Of course, it is not usually necessary to try every dart or even to complete each coding procedure. Since the order is lexicographical, as soon as a letter in the new code-word differs from a letter in the same position in the old one I can terminate the coding; if the new letter is bigger, the old code-word is not maximal and I reject it, and if the new letter is smaller, I try the next dart. If all the darts have been tried and the old code-word hasn't been rejected, I accept (count) it as the representative rooted map of a sensed map. The decoding and coding procedures each run in $O(e)$ time so that the testing procedure runs in $O(e^2)$ time in the worst case: if the map has $2e$ automorphisms, then all the $2e$ codes for this map are identical, so that all the darts must be tried and each code-word must be constructed in its entirety. But, as we will show, the average time for the testing procedure is $O(e \ln e)$. Almost all maps have only the trivial automorphism~\cite{RW95}; so in almost all cases the $2e$ words that code the same map rooted at each of the darts will be distinct. If the old code-word is the ith smallest one among those $2e$ words, this process can be modelled by removing balls at random without replacement from an urn containing $i-1$ black balls (words smaller than the old one) and $2e-i$ white ones (words bigger than the old one) until either a white ball is removed or all the balls are removed. If instead the black balls are replaced, the probability that the next ball will be white decreases, so that the expected number of removals increases. The upper bound thus obtained for the expect number of removals is easy to calculate: it is \begin{equation} \label{eq2} p + 2 (1-p) p + 3 (1-p)^2 p + \ldots = 1/p, \end{equation} where $p$ is the probability of removing a white ball, which is $(2e-i)/(2e-1)$. If $i=2e$, (\ref{eq2}) is not defined, but in this case (when all the balls are black because the old code-word is maximal) (\ref{eq2}) is replaced by the number of removals without replacement, which is $2e$. Since the words coding a given map rooted at all its darts will be generated, $i$ will assume all the values $1, 2, \ldots, 2e$; so the sum of the expected values is less than \begin{equation} \label{eq3} 2e + (2e-1) \, \left[1/1+1/2+1/3+\ldots+1/(2e-1)\right], \end{equation} which is asymptotic to $2 e \ln e$. The expected number of darts that have to be tried for each generated code word is thus $O(\ln e)$. To estimate the cost of comparing an old code-word with a new one, let $i$ be the smallest index of a letter in which the new code-word differs from the old one. The expected value of $i$ is given by (\ref{eq2}), where $p$ is now the probability that two letters chosen at random from an alphabet are distinct, which is equal to $(a‑1)/a$, where $a$ is the number of letters in the alphabet. Since the alphabet has at least two letters if $e > 1$, in the average case the number of letters of the new code-word that have to be constructed is bounded by a constant. However, each coding begins by initializing all the vertices to ``new'', which takes $O(e)$ time; so the average time for testing a code-word for maximality is $O(e \ln e)$. The testing procedure is shown in Figure~\ref{fig1}. To generate the unsensed maps with $e$ edges and $v$ vertices I generate the sensed ones and then, for each sensed map (which I have already constructed by decoding), I reverse the cyclic order of the darts incident to each vertex by constructing the array PREV, where PREV[NEXT[$i$]] = $i$ for each $i$. This step runs in $O(e)$ time. Then I code the reversed map at every dart and compare the new code-word with the old maximal code-word (with the same shortcut, and thus the same average-case time-complexity) and accept the old code-word as the representative rooted map of an unsensed map if none of the comparisons have rejected it. \begin{figure}[htb!] \begin{small} Procedure IsMax ($W$, a code-word of length $2e$ with $p$ parenthesis pairs) \begin{enumerate} \item Decode $W$ into a map $M$ with $e$ edges and $p + 1$ vertices rooted at dart 1 \\ // $O(e)$ time and space; \item Calculate the genus $g$ of $M$; // $O(e)$ time \item Set Maxword to True; \item For $d$ from 2 to $2e$ // $d$ is the current dart \item $\ $ Initialize the coding of $M$ rooted at $d$ by setting all the vertices to new; // $O(e)$ time \item $\ $ For $i$ from 1 to $2e$ \item $\ \ $ Set $X[i]$ to the $i$th letter of the word that codes $M$ rooted at $d$; // $O(1)$ time \item $\ \ $ If $X[i] > W[i]$ then set Maxword to False and exit loop; // $W < X$; so $W$ is not maximal. \item $\ \ $ If $X[i] < W[i]$ then exit loop; // $X < W$; so this dart need no longer be used as a root. \item $\ $ End for $i$; // $O(e)$ worst case and $O(1)$ average-case time \item $\ $ If Maxword = False then exit loop; // $W$ is not maximal; so it will be rejected. \item End for $d$; // $O(e)$ worst-case and $O(\ln e)$ average-case number of iterations \\ // $O(e^2)$ worst-case and $O(e \ln e)$ average-case time to test $W$ for maximality \item If Maxword = True then // $W$ is maximal; so it is chosen as the representative of $M$. $\ $ increase by 1 the number of sensed genus-$g$ maps with $e$ edges and $p+1$ vertices; \end{enumerate} End IsMax. \end{small} \caption{The algorithm for testing whether a code-word represents a sensed map.\label{fig1}} \end{figure} \begin{figure}[htb!] \begin{tabular}{ccc} \begin{minipage}{0.45\textwidth} \begin{center} \begin{tikzpicture}[scale=.4,line cap=round,line join=round,x=1.0cm,y=1.0cm] \node[draw,circle] (v1) at (9,9) {$1$}; \draw (10,10.6) node {$1$}; \draw (10,7.4) node {$8$}; \draw (8.2,10.6) node {$2$}; \draw (8.2,7.4) node {$3$}; \node[draw,circle] (v2) at (5,13) {$2$}; \draw (5.6,11.4) node {$7$}; \node[draw,circle] (v3) at (5,5) {$3$}; \draw (5.6,6.6) node {$6$}; \node[draw,circle] (v4) at (1,1) {$4$}; \draw (4.2,3.4) node {$4$}; \draw (1.6,2.6) node {$5$}; \draw[thick] (v1) to (v2); \draw[thick] (v1) to (v3); \draw[thick] (v3) to (v4); \draw[thick] (v1) to [curve to,looseness=16,in=-40,out=40] (v1); \end{tikzpicture} \end{center} \end{minipage} & & \begin{minipage}{0.45\textwidth} \begin{tabular}{lllllllll} index $i$: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ VERT[$i$]: & 1 & 1 & 1 & 3 & 4 & 3 & 2 & 1 \\ NEXT[$i$]: & 2 & 3 & 8 & 6 & 5 & 4 & 7 & 1 \\ PREV[$i$]: & 8 & 1 & 2 & 6 & 5 & 4 & 7 & 3 \end{tabular} \end{minipage} \end{tabular} \caption{The planar map rooted at dart 1 coded by [ ( ) ( ( ) ) ]. \label{fig2}} \end{figure} For example, at some point during the generation of the rooted planar maps with 4 edges and 4 vertices the word [ ( ) ( ( ) ) ] will be generated. This is the Lehman code-word for the rooted map drawn on the left side of Figure~\ref{fig2}, where the darts and the vertices are labelled in the order in which the edges and vertices are encountered during the decoding procedure (dart 1 is the root). The arrays VERT, NEXT and PREV are shown on the right side of Figure~\ref{fig2}. When this map is coded using any of the darts 2, \ldots, 7 as the root, the first letter is (, which is represented by 0, whereas the first letter of the old code-word is $[$, represented by 2; so the coding terminates immediately. However, when dart 8 is used as the root, the first two letters are [ ]. The second letter of the new code-word is represented by 1, whereas the second letter of the old code-word is represented by 0; so the old code-word is not maximal and is rejected. Later during the generation of the same set of rooted planar maps the word [~]~(~)~(~(~)~) will be generated. This word codes the same map rooted at dart 8. All of the other darts will yield a lexicographically smaller code; so this word will be accepted as the representative of the map drawn in Figure~\ref{fig2} as a sensed map. But when the cyclic orders are reversed and the dart labelled 1 in the diagram is used as the root, the code-word is [~]~(~(~)~)~(~) . This word first differs from the previous one in the fourth letter, which is represented by 0 in the new word and by –1 in the old word; so the old word will be rejected as an unsensed map as soon as the fourth letter has been computed. But when the new word is generated, it will be accepted as both a sensed map and an unsensed map; so this map will count as two sensed maps and one unsensed map. \section{Generating hypermaps} \label{sec4} To generate rooted hypermaps it suffices to generate bicoloured bipartite maps rooted at an edge or, equivalently, rooted at a dart that is incident to a white vertex. This was done by using the following theorem. \begin{theorem} A rooted map is bipartite if and only if its code-word has the property that between every pair of matching brackets or integers there are an odd number of parentheses. \end{theorem} \begin{proof} The spanning tree coded by the parenthesis sub-word is bicoloured, with the vertex incident to the root coloured white. A pair of matching brackets or integers is written when the two darts of an edge $e$ that is not in the spanning tree are encountered during the coding process. Each parenthesis between the members of the pair is written when an edge in the spanning tree is traversed, thus passing from a vertex of one colour to a vertex of the other colour. The two darts of $e$ are thus incident to vertices of opposite colours if and only if the number of parentheses between the matching brackets or integers is odd. If this condition holds for every pair of matching brackets or integers, then the map is properly coloured in two colours and is thus bipartite. If this condition is violated for at least one matching pair, then the map is not properly coloured in two colours, and since the colouring of the spanning tree is uniquely determined by the colour of the vertex containing the root, the map cannot be properly coloured in two colours and is thus not bipartite. This completes the proof. \end{proof} I modified the program to generate just those code words that both avoid the forbidden sub-word and satisfy the condition stated in the theorem, so that it generates the words coding the rooted bipartite maps that are in bijection with hypermaps of the same genus. To this end, I move brackets or integers from right to left, separating the two members of each pair of brackets or integers by an odd number of parentheses, instead of moving parentheses from left to right as in~\cite{Walsh83b}. To test a code word for maximality, I compare it with all the words coding the same map but with a different root incident to a white vertex, and then with all the words coding the orientation-reversed map with any root incident to a white vertex. In this way I generated all the hypermaps -- rooted, sensed, and unsensed -- with up to 12 darts. The time-complexity of the algorithm for generating hypermaps is the same as for maps. Since only the old and the new word have to be stored at any one time and each word is only $O(e)$ letters long, the space-complexity of the generation algorithm is $O(e)$ for both maps and hypermaps. Counting the words and sorting the numbers by genus and the other parameters takes $O(e)$ space for planar maps, $O(e^2)$ space for planar hypermaps and maps that are not necessarily planar, and $O(e^3)$ space for hypermaps that are not necessarily planar. \section{The results of the computation} \label{sec5} The work described in~\cite{Walsh83b} was done in 1979 in the Computing Centre of the USSR Academy of Sciences in Moscow on a BESM-6 computer, which has a 10 megahertz clock speed, and users were restricted to 5 minutes of CPU time per run. Within these limitations I was able to do the calculations for maps with up to only 6 edges, processing a total of 110,410 6-edge rooted maps. I published these results, including a table of numbers of sensed and unsensed maps, in~\cite{Walsh83b}. In 2011, using my Macbook Pro laptop, which has a duo processor and a 2.66 gigahertz clock speed, being subject to no run time restrictions, programming in C instead of FORTRAN and optimizing the algorithms, I was able to extend the calculations up to 11 edges; the run time for 11 edges, which processes 285,764,591,114 rooted maps, was about a week. For 10 edges it was about a day, for 9 edges about three hours, for 8 edges about 20 minutes, for 7 edges about 2 minutes, for 6 edges about 10 seconds and for fewer than 6 edges it was too short to be measured. In each case the time was roughly proportional to the number of rooted maps, verifying experimentally the above average-case time complexity for maximality testing. Further verification was provided by the following time trial: it took two minutes to generate all the rooted planar maps with 10 edges and less than six minutes to generate all the unsensed planar maps with 10 edges. For unsensed hypermaps, the computation time was 8 seconds for 9 darts, 2 minutes for 10 darts, 33 minutes for 11 darts and about 10 hours (to process 5,201,061,455 rooted bipartite maps) for 12 darts. The numbers of rooted maps generated by my program agree with the tables in my joint paper with Prof. Lehman~\cite{W1}; these tables go up to 11 edges, and the tables in~\cite{Walsh71} go up to 14 edges. The numbers of sensed maps agree with the numbers calculated jointly with Giorgetti and Mednykh without generating maps; tables for the non-planar maps with up to 11 edges appear in~\cite{WGMunrooted} and~\cite{Walsh13}. The numbers of unsensed maps with up to 6 edges agree with the tables in~\cite{Walsh83b}. The numbers of unsensed planar maps agree with those in unpublished tables given to me by Wormald, who counted those maps and published his results in~\cite{Wormald81a} and~\cite{Wormald81b}. The numbers of rooted and sensed hypermaps of genus 0 and 1 with $d$ darts agree with those published by Mednykh and Nedela~\cite{MN10}. The numbers of rooted hypermaps of genus 0 and 1, counted by number of vertices, edges and faces, agree with those published by Chauve~\cite{Chauve03} and Arqu\`es~\cite{Arq87}, respectively. For unsensed non-planar maps with more than 6 edges, as well as for all the other types of hypermaps, the numbers I generated are, as far as I know, new. The source code is available as a text file~\cite{WalshCode}. It will run on any 64-bit computer that runs C programs. \section{Acknowledgment} I wish to thank NSERC for partially supporting this research, and Alain Giorgetti and Alexander Mednykh for suggestions for improving the presentation of this article. \appendix \subsubsection*{Appendix A: The number of unsensed genus-g maps with e edges and v vertices.} \label{appA} \begin{scriptsize} \begin{verbatim} E v g=0 g=1 g=2 g=3 g=4 g=5 all g 0 1 1 1 0 sum 1 1 1 1 1 1 1 2 1 1 1 sum 2 2 2 1 1 1 2 2 2 2 0 2 2 3 1 0 1 2 sum 4 1 5 3 1 2 3 5 3 2 5 3 8 3 3 5 0 5 3 4 2 0 2 3 sum 14 6 20 4 1 3 10 4 17 4 2 13 20 0 33 4 3 20 10 0 30 4 4 13 0 0 13 4 5 3 0 0 3 4 sum 52 40 4 96 5 1 6 35 38 79 5 2 35 125 38 198 5 3 83 125 0 208 5 4 83 35 0 118 5 5 35 0 0 35 5 6 6 0 0 6 5 sum 248 320 76 644 6 1 12 132 328 82 554 6 2 104 728 739 0 1571 6 3 340 1226 328 0 1894 6 4 504 728 0 0 1232 6 5 340 132 0 0 472 6 6 104 0 0 0 104 6 7 12 0 0 0 12 6 sum 1416 2946 1395 82 5839 7 1 27 513 2569 2174 5283 7 2 315 4036 9906 2174 16431 7 3 1401 10133 9906 0 21440 7 4 2843 10133 2569 0 15545 7 5 2843 4036 0 0 6879 7 6 1401 513 0 0 1914 7 7 315 0 0 0 315 7 8 27 0 0 0 27 7 sum 9172 29364 24950 4348 67834 8 1 65 2072 18512 37439 7258 65346 8 2 1021 21733 105905 85172 0 213831 8 3 5809 75202 178502 37439 0 296952 8 4 15578 111544 105905 0 0 233027 8 5 21420 75202 18512 0 0 115134 8 6 15578 21733 0 0 0 37311 8 7 5809 2072 0 0 0 7881 8 8 1021 0 0 0 0 1021 8 9 65 0 0 0 0 65 8 sum 66366 309558 427336 160050 7258 970568 9 1 175 8558 124044 488891 344488 966156 9 2 3407 113721 967844 1859361 344488 3288821 9 3 24299 514014 2401662 1859361 0 4799336 9 4 82546 1046261 2401662 488891 0 4019360 9 5 149007 1046261 967844 0 0 2163112 9 6 149007 514014 124044 0 0 787065 9 7 82546 113721 0 0 0 196267 9 8 24299 8558 0 0 0 32857 9 9 3407 0 0 0 0 3407 9 10 175 0 0 0 0 175 9 sum 518868 3365108 6987100 4696504 688976 16256556 10 1 490 35655 781919 5293283 8808724 1491629 16411700 10 2 11814 580810 7887415 29372094 19848849 0 57700982 10 3 102010 3294692 26625471 49022864 8808724 0 87853761 10 4 426879 8728573 39172217 29372094 0 0 77699763 10 5 972660 11966785 26625471 5293283 0 0 44858199 10 6 1273644 8728573 7887415 0 0 0 17889632 10 7 972660 3294692 781919 0 0 0 5049271 10 8 426879 580810 0 0 0 0 1007689 10 9 102010 35655 0 0 0 0 137665 10 10 11814 0 0 0 0 0 11814 10 11 490 0 0 0 0 0 490 10 sum 4301350 37246245 109761827 118353618 37466297 1491629 308620966 11 1 1473 149257 4690016 50026987 159968175 97864389 312700297 11 2 41893 2901436 58891739 374871812 596357213 97864389 1130928482 11 3 429509 20057276 256786053 912749995 596357213 0 1786380046 11 4 2158241 66570286 513820635 912749995 159968175 0 1655267332 11 5 6030752 118697249 513820635 374871812 0 0 1013420448 11 6 9953314 118697249 256786053 50026987 0 0 435463603 11 7 9953314 66570286 58891739 0 0 0 135415339 11 8 6030752 20057276 4690016 0 0 0 30778044 11 9 2158241 2901436 0 0 0 0 5059677 11 10 429509 149257 0 0 0 0 578766 11 11 41893 0 0 0 0 0 41893 11 12 1473 0 0 0 0 0 1473 11 sum 37230364 416751008 1668376886 2675297588 1512650776 195728778 6506035400 \end{verbatim} \end{scriptsize} \subsubsection*{Appendix B: The number of hypermaps of genus g.} \label{appB} \begin{scriptsize} \begin{verbatim} Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 all g 1 1 1 1 1 1 sum 1 1 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 all g 1 1 1 1 1 1 sum 1 1 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 all g 1 1 1 1 1 1 sum 1 1 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 all g 2 1 1 1 1 2 1 2 1 1 2 2 1 1 1 2 sum 3 3 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 all g 2 1 1 1 1 2 1 2 1 1 2 2 1 1 1 2 sum 3 3 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 all g 2 1 1 1 1 2 1 2 1 1 2 2 1 1 1 2 sum 3 3 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 all g 3 1 1 1 1 2 3 1 2 3 0 3 3 2 1 3 0 3 3 1 3 1 0 1 3 2 2 3 0 3 3 3 1 1 0 1 3 sum 12 1 13 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 all g 3 1 1 1 1 2 3 1 2 1 0 1 3 2 1 1 0 1 3 1 3 1 0 1 3 2 2 1 0 1 3 3 1 1 0 1 3 sum 6 1 7 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 all g 3 1 1 1 1 2 3 1 2 1 0 1 3 2 1 1 0 1 3 1 3 1 0 1 3 2 2 1 0 1 3 3 1 1 0 1 3 sum 6 1 7 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 all g 4 1 1 1 5 6 4 1 2 6 5 11 4 2 1 6 5 11 4 1 3 6 0 6 4 2 2 17 0 17 4 3 1 6 0 6 4 1 4 1 0 1 4 2 3 6 0 6 4 3 2 6 0 6 4 4 1 1 0 1 4 sum 56 15 71 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 all g 4 1 1 1 2 3 4 1 2 2 2 4 4 2 1 2 2 4 4 1 3 2 0 2 4 2 2 5 0 5 4 3 1 2 0 2 4 1 4 1 0 1 4 2 3 2 0 2 4 3 2 2 0 2 4 4 1 1 0 1 4 sum 20 6 26 \end{verbatim} \end{scriptsize} \newpage \begin{scriptsize} \begin{verbatim} Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 all g 4 1 1 1 2 3 4 1 2 2 2 4 4 2 1 2 2 4 4 1 3 2 0 2 4 2 2 5 0 5 4 3 1 2 0 2 4 1 4 1 0 1 4 2 3 2 0 2 4 3 2 2 0 2 4 4 1 1 0 1 4 sum 20 6 26 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 all g 5 1 1 1 15 8 24 5 1 2 10 40 0 50 5 2 1 10 40 0 50 5 1 3 20 15 0 35 5 2 2 55 40 0 95 5 3 1 20 15 0 35 5 1 4 10 0 0 10 5 2 3 55 0 0 55 5 3 2 55 0 0 55 5 4 1 10 0 0 10 5 1 5 1 0 0 1 5 2 4 10 0 0 10 5 3 3 20 0 0 20 5 4 2 10 0 0 10 5 5 1 1 0 0 1 5 sum 288 165 8 461 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 all g 5 1 1 1 3 4 8 5 1 2 2 8 0 10 5 2 1 2 8 0 10 5 1 3 4 3 0 7 5 2 2 11 8 0 19 5 3 1 4 3 0 7 5 1 4 2 0 0 2 5 2 3 11 0 0 11 5 3 2 11 0 0 11 5 4 1 2 0 0 2 5 1 5 1 0 0 1 5 2 4 2 0 0 2 5 3 3 4 0 0 4 5 4 2 2 0 0 2 5 5 1 1 0 0 1 5 sum 60 33 4 97 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 all g 5 1 1 1 3 4 8 5 1 2 2 7 0 9 5 2 1 2 7 0 9 5 1 3 4 3 0 7 5 2 2 10 7 0 17 5 3 1 4 3 0 7 5 1 4 2 0 0 2 5 2 3 10 0 0 10 5 3 2 10 0 0 10 5 4 1 2 0 0 2 5 1 5 1 0 0 1 5 2 4 2 0 0 2 5 3 3 4 0 0 4 5 4 2 2 0 0 2 5 5 1 1 0 0 1 5 sum 57 30 4 91 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 all g 6 1 1 1 35 84 120 6 1 2 15 175 84 274 6 2 1 15 175 84 274 6 1 3 50 175 0 225 6 2 2 135 456 0 591 6 3 1 50 175 0 225 6 1 4 50 35 0 85 6 2 3 262 175 0 437 6 3 2 262 175 0 437 6 4 1 50 35 0 85 6 1 5 15 0 0 15 6 2 4 135 0 0 135 6 3 3 262 0 0 262 6 4 2 135 0 0 135 6 5 1 15 0 0 15 6 1 6 1 0 0 1 6 2 5 15 0 0 15 6 3 4 50 0 0 50 6 4 3 50 0 0 50 6 5 2 15 0 0 15 6 6 1 1 0 0 1 6 sum 1584 1611 252 3447 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 all g 6 1 1 1 7 16 24 6 1 2 3 31 16 50 6 2 1 3 31 16 50 6 1 3 10 31 0 41 6 2 2 24 78 0 102 6 3 1 10 31 0 41 6 1 4 10 7 0 17 6 2 3 46 31 0 77 6 3 2 46 31 0 77 6 4 1 10 7 0 17 6 1 5 3 0 0 3 6 2 4 24 0 0 24 6 3 3 46 0 0 46 6 4 2 24 0 0 24 6 5 1 3 0 0 3 6 1 6 1 0 0 1 6 2 5 3 0 0 3 6 3 4 10 0 0 10 6 4 3 10 0 0 10 6 5 2 3 0 0 3 6 6 1 1 0 0 1 6 sum 291 285 48 624 \end{verbatim} \end{scriptsize} \newpage \begin{scriptsize} \begin{verbatim} Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 all g 6 1 1 1 6 13 20 6 1 2 3 22 13 38 6 2 1 3 22 13 38 6 1 3 8 22 0 30 6 2 2 21 61 0 82 6 3 1 8 22 0 30 6 1 4 8 6 0 14 6 2 3 36 22 0 58 6 3 2 36 22 0 58 6 4 1 8 6 0 14 6 1 5 3 0 0 3 6 2 4 21 0 0 21 6 3 3 36 0 0 36 6 4 2 21 0 0 21 6 5 1 3 0 0 3 6 1 6 1 0 0 1 6 2 5 3 0 0 3 6 3 4 8 0 0 8 6 4 3 8 0 0 8 6 5 2 3 0 0 3 6 6 1 1 0 0 1 6 sum 240 211 39 490 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 all g 7 1 1 1 70 469 180 720 7 1 2 21 560 1183 0 1764 7 2 1 21 560 1183 0 1764 7 1 3 105 1050 469 0 1624 7 2 2 280 2695 1183 0 4158 7 3 1 105 1050 469 0 1624 7 1 4 175 560 0 0 735 7 2 3 889 2695 0 0 3584 7 3 2 889 2695 0 0 3584 7 4 1 175 560 0 0 735 7 1 5 105 70 0 0 175 7 2 4 889 560 0 0 1449 7 3 3 1694 1050 0 0 2744 7 4 2 889 560 0 0 1449 7 5 1 105 70 0 0 175 7 1 6 21 0 0 0 21 7 2 5 280 0 0 0 280 7 3 4 889 0 0 0 889 7 4 3 889 0 0 0 889 7 5 2 280 0 0 0 280 7 6 1 21 0 0 0 21 7 1 7 1 0 0 0 1 7 2 6 21 0 0 0 21 7 3 5 105 0 0 0 105 7 4 4 175 0 0 0 175 7 5 3 105 0 0 0 105 7 6 2 21 0 0 0 21 7 7 1 1 0 0 0 1 7 sum 9152 14805 4956 180 29093 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 all g 7 1 1 1 10 67 30 108 7 1 2 3 80 169 0 252 7 2 1 3 80 169 0 252 7 1 3 15 150 67 0 232 7 2 2 40 385 169 0 594 7 3 1 15 150 67 0 232 7 1 4 25 80 0 0 105 7 2 3 127 385 0 0 512 7 3 2 127 385 0 0 512 7 4 1 25 80 0 0 105 7 1 5 15 10 0 0 25 7 2 4 127 80 0 0 207 7 3 3 242 150 0 0 392 7 4 2 127 80 0 0 207 7 5 1 15 10 0 0 25 7 1 6 3 0 0 0 3 7 2 5 40 0 0 0 40 7 3 4 127 0 0 0 127 7 4 3 127 0 0 0 127 7 5 2 40 0 0 0 40 7 6 1 3 0 0 0 3 7 1 7 1 0 0 0 1 7 2 6 3 0 0 0 3 7 3 5 15 0 0 0 15 7 4 4 25 0 0 0 25 7 5 3 15 0 0 0 15 7 6 2 3 0 0 0 3 7 7 1 1 0 0 0 1 7 sum 1310 2115 708 30 4163 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 all g 7 1 1 1 8 44 25 78 7 1 2 3 51 108 0 162 7 2 1 3 51 108 0 162 7 1 3 12 91 44 0 147 7 2 2 33 249 108 0 390 7 3 1 12 91 44 0 147 7 1 4 17 51 0 0 68 7 2 3 90 249 0 0 339 7 3 2 90 249 0 0 339 7 4 1 17 51 0 0 68 7 1 5 12 8 0 0 20 7 2 4 90 51 0 0 141 7 3 3 171 91 0 0 262 7 4 2 90 51 0 0 141 7 5 1 12 8 0 0 20 7 1 6 3 0 0 0 3 7 2 5 33 0 0 0 33 7 3 4 90 0 0 0 90 7 4 3 90 0 0 0 90 7 5 2 33 0 0 0 33 7 6 1 3 0 0 0 3 7 1 7 1 0 0 0 1 7 2 6 3 0 0 0 3 7 3 5 12 0 0 0 12 7 4 4 17 0 0 0 17 7 5 3 12 0 0 0 12 7 6 2 3 0 0 0 3 7 7 1 1 0 0 0 1 7 sum 954 1350 456 25 2785 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 all g 8 1 1 1 126 1869 3044 5040 8 1 2 28 1470 8526 3044 13068 8 2 1 28 1470 8526 3044 13068 8 1 3 196 4410 8526 0 13132 8 2 2 518 11199 21229 0 32946 8 3 1 196 4410 8526 0 13132 8 1 4 490 4410 1869 0 6769 8 2 3 2436 20684 8526 0 31646 8 3 2 2436 20684 8526 0 31646 8 4 1 490 4410 1869 0 6769 8 1 5 490 1470 0 0 1960 8 2 4 3985 11199 0 0 15184 8 3 3 7500 20684 0 0 28184 8 4 2 3985 11199 0 0 15184 8 5 1 490 1470 0 0 1960 8 1 6 196 126 0 0 322 8 2 5 2436 1470 0 0 3906 8 3 4 7500 4410 0 0 11910 8 4 3 7500 4410 0 0 11910 8 5 2 2436 1470 0 0 3906 8 6 1 196 126 0 0 322 8 1 7 28 0 0 0 28 8 2 6 518 0 0 0 518 8 3 5 2436 0 0 0 2436 8 4 4 3985 0 0 0 3985 8 5 3 2436 0 0 0 2436 8 6 2 518 0 0 0 518 8 7 1 28 0 0 0 28 8 1 8 1 0 0 0 1 8 2 7 28 0 0 0 28 8 3 6 196 0 0 0 196 8 4 5 490 0 0 0 490 8 5 4 490 0 0 0 490 8 6 3 196 0 0 0 196 8 7 2 28 0 0 0 28 8 8 1 1 0 0 0 1 8 sum 54912 131307 77992 9132 273343 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 all g 8 1 1 1 17 237 385 640 8 1 2 4 187 1072 385 1648 8 2 1 4 187 1072 385 1648 8 1 3 26 557 1072 0 1655 8 2 2 67 1409 2664 0 4140 8 3 1 26 557 1072 0 1655 8 1 4 64 557 237 0 858 8 2 3 309 2597 1072 0 3978 8 3 2 309 2597 1072 0 3978 8 4 1 64 557 237 0 858 8 1 5 64 187 0 0 251 8 2 4 505 1409 0 0 1914 8 3 3 946 2597 0 0 3543 8 4 2 505 1409 0 0 1914 8 5 1 64 187 0 0 251 8 1 6 26 17 0 0 43 8 2 5 309 187 0 0 496 8 3 4 946 557 0 0 1503 8 4 3 946 557 0 0 1503 8 5 2 309 187 0 0 496 8 6 1 26 17 0 0 43 8 1 7 4 0 0 0 4 8 2 6 67 0 0 0 67 8 3 5 309 0 0 0 309 8 4 4 505 0 0 0 505 8 5 3 309 0 0 0 309 8 6 2 67 0 0 0 67 8 7 1 4 0 0 0 4 8 1 8 1 0 0 0 1 8 2 7 4 0 0 0 4 8 3 6 26 0 0 0 26 8 4 5 64 0 0 0 64 8 5 4 64 0 0 0 64 8 6 3 26 0 0 0 26 8 7 2 4 0 0 0 4 8 8 1 1 0 0 0 1 8 sum 6975 16533 9807 1155 34470 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 all g 8 1 1 1 13 140 226 380 8 1 2 4 112 596 226 938 8 2 1 4 112 596 226 938 8 1 3 19 314 596 0 929 8 2 2 54 840 1558 0 2452 8 3 1 19 314 596 0 929 8 1 4 41 314 140 0 495 8 2 3 205 1507 596 0 2308 8 3 2 205 1507 596 0 2308 8 4 1 41 314 140 0 495 8 1 5 41 112 0 0 153 8 2 4 325 840 0 0 1165 8 3 3 604 1507 0 0 2111 8 4 2 325 840 0 0 1165 8 5 1 41 112 0 0 153 8 1 6 19 13 0 0 32 8 2 5 205 112 0 0 317 8 3 4 604 314 0 0 918 8 4 3 604 314 0 0 918 8 5 2 205 112 0 0 317 8 6 1 19 13 0 0 32 8 1 7 4 0 0 0 4 8 2 6 54 0 0 0 54 8 3 5 205 0 0 0 205 8 4 4 325 0 0 0 325 8 5 3 205 0 0 0 205 8 6 2 54 0 0 0 54 8 7 1 4 0 0 0 4 8 1 8 1 0 0 0 1 8 2 7 4 0 0 0 4 8 3 6 19 0 0 0 19 8 4 5 41 0 0 0 41 8 5 4 41 0 0 0 41 8 6 3 19 0 0 0 19 8 7 2 4 0 0 0 4 8 8 1 1 0 0 0 1 8 sum 4566 9636 5554 678 20434 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 all g 9 1 1 1 210 5985 26060 8064 40320 9 1 2 36 3360 42588 63600 0 109584 9 2 1 36 3360 42588 63600 0 109584 9 1 3 336 14700 77028 26060 0 118124 9 2 2 882 37035 189999 63600 0 291516 9 3 1 336 14700 77028 26060 0 118124 9 1 4 1176 23520 42588 0 0 67284 9 2 3 5754 108285 189999 0 0 304038 9 3 2 5754 108285 189999 0 0 304038 9 4 1 1176 23520 42588 0 0 67284 9 1 5 1764 14700 5985 0 0 22449 9 2 4 13941 108285 42588 0 0 164814 9 3 3 26004 197896 77028 0 0 300928 9 4 2 13941 108285 42588 0 0 164814 9 5 1 1764 14700 5985 0 0 22449 9 1 6 1176 3360 0 0 0 4536 9 2 5 13941 37035 0 0 0 50976 9 3 4 42015 108285 0 0 0 150300 9 4 3 42015 108285 0 0 0 150300 9 5 2 13941 37035 0 0 0 50976 9 6 1 1176 3360 0 0 0 4536 9 1 7 336 210 0 0 0 546 9 2 6 5754 3360 0 0 0 9114 9 3 5 26004 14700 0 0 0 40704 9 4 4 42015 23520 0 0 0 65535 9 5 3 26004 14700 0 0 0 40704 9 6 2 5754 3360 0 0 0 9114 9 7 1 336 210 0 0 0 546 9 1 8 36 0 0 0 0 36 9 2 7 882 0 0 0 0 882 9 3 6 5754 0 0 0 0 5754 9 4 5 13941 0 0 0 0 13941 9 5 4 13941 0 0 0 0 13941 9 6 3 5754 0 0 0 0 5754 9 7 2 882 0 0 0 0 882 9 8 1 36 0 0 0 0 36 9 1 9 1 0 0 0 0 1 9 2 8 36 0 0 0 0 36 9 3 7 336 0 0 0 0 336 9 4 6 1176 0 0 0 0 1176 9 5 5 1764 0 0 0 0 1764 9 6 4 1176 0 0 0 0 1176 9 7 3 336 0 0 0 0 336 9 8 2 36 0 0 0 0 36 9 9 1 1 0 0 0 0 1 9 sum 339456 1138261 1074564 268980 8064 2829325 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 all g 9 1 1 1 24 667 2900 900 4492 9 1 2 4 374 4736 7070 0 12184 9 2 1 4 374 4736 7070 0 12184 9 1 3 38 1634 8560 2900 0 13132 9 2 2 98 4115 21113 7070 0 32396 9 3 1 38 1634 8560 2900 0 13132 9 1 4 132 2616 4736 0 0 7484 9 2 3 640 12033 21113 0 0 33786 9 3 2 640 12033 21113 0 0 33786 9 4 1 132 2616 4736 0 0 7484 9 1 5 196 1634 667 0 0 2497 9 2 4 1549 12033 4736 0 0 18318 9 3 3 2890 21990 8560 0 0 33440 9 4 2 1549 12033 4736 0 0 18318 9 5 1 196 1634 667 0 0 2497 9 1 6 132 374 0 0 0 506 9 2 5 1549 4115 0 0 0 5664 9 3 4 4671 12033 0 0 0 16704 9 4 3 4671 12033 0 0 0 16704 9 5 2 1549 4115 0 0 0 5664 9 6 1 132 374 0 0 0 506 9 1 7 38 24 0 0 0 62 9 2 6 640 374 0 0 0 1014 9 3 5 2890 1634 0 0 0 4524 9 4 4 4671 2616 0 0 0 7287 9 5 3 2890 1634 0 0 0 4524 9 6 2 640 374 0 0 0 1014 9 7 1 38 24 0 0 0 62 9 1 8 4 0 0 0 0 4 9 2 7 98 0 0 0 0 98 9 3 6 640 0 0 0 0 640 9 4 5 1549 0 0 0 0 1549 9 5 4 1549 0 0 0 0 1549 9 6 3 640 0 0 0 0 640 9 7 2 98 0 0 0 0 98 9 8 1 4 0 0 0 0 4 9 1 9 1 0 0 0 0 1 9 2 8 4 0 0 0 0 4 9 3 7 38 0 0 0 0 38 9 4 6 132 0 0 0 0 132 9 5 5 196 0 0 0 0 196 9 6 4 132 0 0 0 0 132 9 7 3 38 0 0 0 0 38 9 8 2 4 0 0 0 0 4 9 9 1 1 0 0 0 0 1 9 sum 37746 126501 119436 29910 900 314493 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 all g 9 1 1 1 17 366 1530 524 2438 9 1 2 4 213 2500 3759 0 6476 9 2 1 4 213 2500 3759 0 6476 9 1 3 27 879 4474 1530 0 6910 9 2 2 76 2309 11286 3759 0 17430 9 3 1 27 879 4474 1530 0 6910 9 1 4 78 1388 2500 0 0 3966 9 2 3 403 6568 11286 0 0 18257 9 3 2 403 6568 11286 0 0 18257 9 4 1 78 1388 2500 0 0 3966 9 1 5 116 879 366 0 0 1361 9 2 4 920 6568 2500 0 0 9988 9 3 3 1743 12067 4474 0 0 18284 9 4 2 920 6568 2500 0 0 9988 9 5 1 116 879 366 0 0 1361 9 1 6 78 213 0 0 0 291 9 2 5 920 2309 0 0 0 3229 9 3 4 2747 6568 0 0 0 9315 9 4 3 2747 6568 0 0 0 9315 9 5 2 920 2309 0 0 0 3229 9 6 1 78 213 0 0 0 291 9 1 7 27 17 0 0 0 44 9 2 6 403 213 0 0 0 616 9 3 5 1743 879 0 0 0 2622 9 4 4 2747 1388 0 0 0 4135 9 5 3 1743 879 0 0 0 2622 9 6 2 403 213 0 0 0 616 9 7 1 27 17 0 0 0 44 9 1 8 4 0 0 0 0 4 9 2 7 76 0 0 0 0 76 9 3 6 403 0 0 0 0 403 9 4 5 920 0 0 0 0 920 9 5 4 920 0 0 0 0 920 9 6 3 403 0 0 0 0 403 9 7 2 76 0 0 0 0 76 9 8 1 4 0 0 0 0 4 9 1 9 1 0 0 0 0 1 9 2 8 4 0 0 0 0 4 9 3 7 27 0 0 0 0 27 9 4 6 78 0 0 0 0 78 9 5 5 116 0 0 0 0 116 9 6 4 78 0 0 0 0 78 9 7 3 27 0 0 0 0 27 9 8 2 4 0 0 0 0 4 9 9 1 1 0 0 0 0 1 9 sum 22641 69169 63378 15867 524 171579 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 all g 10 1 1 1 330 16401 152900 193248 362880 10 1 2 45 6930 167013 659340 193248 1026576 10 2 1 45 6930 167013 659340 193248 1026576 10 1 3 540 41580 471240 659340 0 1172700 10 2 2 1410 104115 1154095 1595480 0 2855100 10 3 1 540 41580 471240 659340 0 1172700 10 1 4 2520 97020 471240 152900 0 723680 10 2 3 12180 440440 2068070 659340 0 3180030 10 3 2 12180 440440 2068070 659340 0 3180030 10 4 1 2520 97020 471240 152900 0 723680 10 1 5 5292 97020 167013 0 0 269325 10 2 4 40935 697250 1154095 0 0 1892280 10 3 3 75840 1264310 2068070 0 0 3408220 10 4 2 40935 697250 1154095 0 0 1892280 10 5 1 5292 97020 167013 0 0 269325 10 1 6 5292 41580 16401 0 0 63273 10 2 5 60626 440440 167013 0 0 668079 10 3 4 179860 1264310 471240 0 0 1915410 10 4 3 179860 1264310 471240 0 0 1915410 10 5 2 60626 440440 167013 0 0 668079 10 6 1 5292 41580 16401 0 0 63273 10 1 7 2520 6930 0 0 0 9450 10 2 6 40935 104115 0 0 0 145050 10 3 5 179860 440440 0 0 0 620300 10 4 4 288025 697250 0 0 0 985275 10 5 3 179860 440440 0 0 0 620300 10 6 2 40935 104115 0 0 0 145050 10 7 1 2520 6930 0 0 0 9450 10 1 8 540 330 0 0 0 870 10 2 7 12180 6930 0 0 0 19110 10 3 6 75840 41580 0 0 0 117420 10 4 5 179860 97020 0 0 0 276880 10 5 4 179860 97020 0 0 0 276880 10 6 3 75840 41580 0 0 0 117420 10 7 2 12180 6930 0 0 0 19110 10 8 1 540 330 0 0 0 870 10 1 9 45 0 0 0 0 45 10 2 8 1410 0 0 0 0 1410 10 3 7 12180 0 0 0 0 12180 10 4 6 40935 0 0 0 0 40935 10 5 5 60626 0 0 0 0 60626 10 6 4 40935 0 0 0 0 40935 10 7 3 12180 0 0 0 0 12180 10 8 2 1410 0 0 0 0 1410 10 9 1 45 0 0 0 0 45 10 1 10 1 0 0 0 0 1 10 2 9 45 0 0 0 0 45 10 3 8 540 0 0 0 0 540 10 4 7 2520 0 0 0 0 2520 10 5 6 5292 0 0 0 0 5292 10 6 5 5292 0 0 0 0 5292 10 7 4 2520 0 0 0 0 2520 10 8 3 540 0 0 0 0 540 10 9 2 45 0 0 0 0 45 10 10 1 1 0 0 0 0 1 10 sum 2149888 9713835 13545216 6010220 579744 31998903 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 all g 10 1 1 1 34 1649 15308 19344 36336 10 1 2 5 698 16725 65972 19344 102744 10 2 1 5 698 16725 65972 19344 102744 10 1 3 56 4172 47164 65972 0 117364 10 2 2 144 10434 115478 159608 0 285664 10 3 1 56 4172 47164 65972 0 117364 10 1 4 256 9724 47164 15308 0 72452 10 2 3 1226 44091 206895 65972 0 318184 10 3 2 1226 44091 206895 65972 0 318184 10 4 1 256 9724 47164 15308 0 72452 10 1 5 536 9724 16725 0 0 26985 10 2 4 4111 69790 115478 0 0 189379 10 3 3 7606 126519 206895 0 0 341020 10 4 2 4111 69790 115478 0 0 189379 10 5 1 536 9724 16725 0 0 26985 10 1 6 536 4172 1649 0 0 6357 10 2 5 6081 44091 16725 0 0 66897 10 3 4 18019 126519 47164 0 0 191702 10 4 3 18019 126519 47164 0 0 191702 10 5 2 6081 44091 16725 0 0 66897 10 6 1 536 4172 1649 0 0 6357 10 1 7 256 698 0 0 0 954 10 2 6 4111 10434 0 0 0 14545 10 3 5 18019 44091 0 0 0 62110 10 4 4 28852 69790 0 0 0 98642 10 5 3 18019 44091 0 0 0 62110 10 6 2 4111 10434 0 0 0 14545 10 7 1 256 698 0 0 0 954 10 1 8 56 34 0 0 0 90 10 2 7 1226 698 0 0 0 1924 10 3 6 7606 4172 0 0 0 11778 10 4 5 18019 9724 0 0 0 27743 10 5 4 18019 9724 0 0 0 27743 10 6 3 7606 4172 0 0 0 11778 10 7 2 1226 698 0 0 0 1924 10 8 1 56 34 0 0 0 90 10 1 9 5 0 0 0 0 5 10 2 8 144 0 0 0 0 144 10 3 7 1226 0 0 0 0 1226 10 4 6 4111 0 0 0 0 4111 10 5 5 6081 0 0 0 0 6081 10 6 4 4111 0 0 0 0 4111 10 7 3 1226 0 0 0 0 1226 10 8 2 144 0 0 0 0 144 10 9 1 5 0 0 0 0 5 10 1 10 1 0 0 0 0 1 10 2 9 5 0 0 0 0 5 10 3 8 56 0 0 0 0 56 10 4 7 256 0 0 0 0 256 10 5 6 536 0 0 0 0 536 10 6 5 536 0 0 0 0 536 10 7 4 256 0 0 0 0 256 10 8 3 56 0 0 0 0 56 10 9 2 5 0 0 0 0 5 10 10 1 1 0 0 0 0 1 10 sum 215602 972441 1355400 601364 58032 3202839 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 all g 10 1 1 1 24 883 7866 9970 18744 10 1 2 5 388 8622 33635 9970 52620 10 2 1 5 388 8622 33635 9970 52620 10 1 3 38 2196 24085 33635 0 59954 10 2 2 110 5676 59772 82472 0 148030 10 3 1 38 2196 24085 33635 0 59954 10 1 4 148 5037 24085 7866 0 37136 10 2 3 746 23303 106787 33635 0 164471 10 3 2 746 23303 106787 33635 0 164471 10 4 1 148 5037 24085 7866 0 37136 10 1 5 298 5037 8622 0 0 13957 10 2 4 2344 36669 59772 0 0 98785 10 3 3 4386 66787 106787 0 0 177960 10 4 2 2344 36669 59772 0 0 98785 10 5 1 298 5037 8622 0 0 13957 10 1 6 298 2196 883 0 0 3377 10 2 5 3391 23303 8622 0 0 35316 10 3 4 10097 66787 24085 0 0 100969 10 4 3 10097 66787 24085 0 0 100969 10 5 2 3391 23303 8622 0 0 35316 10 6 1 298 2196 883 0 0 3377 10 1 7 148 388 0 0 0 536 10 2 6 2344 5676 0 0 0 8020 10 3 5 10097 23303 0 0 0 33400 10 4 4 16103 36669 0 0 0 52772 10 5 3 10097 23303 0 0 0 33400 10 6 2 2344 5676 0 0 0 8020 10 7 1 148 388 0 0 0 536 10 1 8 38 24 0 0 0 62 10 2 7 746 388 0 0 0 1134 10 3 6 4386 2196 0 0 0 6582 10 4 5 10097 5037 0 0 0 15134 10 5 4 10097 5037 0 0 0 15134 10 6 3 4386 2196 0 0 0 6582 10 7 2 746 388 0 0 0 1134 10 8 1 38 24 0 0 0 62 10 1 9 5 0 0 0 0 5 10 2 8 110 0 0 0 0 110 10 3 7 746 0 0 0 0 746 10 4 6 2344 0 0 0 0 2344 10 5 5 3391 0 0 0 0 3391 10 6 4 2344 0 0 0 0 2344 10 7 3 746 0 0 0 0 746 10 8 2 110 0 0 0 0 110 10 9 1 5 0 0 0 0 5 10 1 10 1 0 0 0 0 1 10 2 9 5 0 0 0 0 5 10 3 8 38 0 0 0 0 38 10 4 7 148 0 0 0 0 148 10 5 6 298 0 0 0 0 298 10 6 5 298 0 0 0 0 298 10 7 4 148 0 0 0 0 148 10 8 3 38 0 0 0 0 38 10 9 2 5 0 0 0 0 5 10 10 1 1 0 0 0 0 1 10 sum 121823 513012 698568 307880 29910 1671193 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 g=5 all g 11 1 1 1 495 39963 696905 2286636 604800 3628800 11 1 2 55 13200 550011 4606910 5458464 0 10628640 11 2 1 55 13200 550011 4606910 5458464 0 10628640 11 1 3 825 103950 2221065 8141100 2286636 0 12753576 11 2 2 2145 259017 5409019 19571123 5458464 0 30699768 11 3 1 825 103950 2221065 8141100 2286636 0 12753576 11 1 4 4950 332640 3465000 4606910 0 0 8409500 11 2 3 23694 1493525 15014846 19571123 0 0 36103188 11 3 2 23694 1493525 15014846 19571123 0 0 36103188 11 4 1 4950 332640 3465000 4606910 0 0 8409500 11 1 5 13860 485100 2221065 696905 0 0 3416930 11 2 4 105435 3420835 15014846 4606910 0 0 23148026 11 3 3 194304 6165478 26717482 8141100 0 0 41218364 11 4 2 105435 3420835 15014846 4606910 0 0 23148026 11 5 1 13860 485100 2221065 696905 0 0 3416930 11 1 6 19404 332640 550011 0 0 0 902055 11 2 5 216601 3420835 5409019 0 0 0 9046455 11 3 4 634865 9684433 15014846 0 0 0 25334144 11 4 3 634865 9684433 15014846 0 0 0 25334144 11 5 2 216601 3420835 5409019 0 0 0 9046455 11 6 1 19404 332640 550011 0 0 0 902055 11 1 7 13860 103950 39963 0 0 0 157773 11 2 6 216601 1493525 550011 0 0 0 2260137 11 3 5 931854 6165478 2221065 0 0 0 9318397 11 4 4 1482250 9684433 3465000 0 0 0 14631683 11 5 3 931854 6165478 2221065 0 0 0 9318397 11 6 2 216601 1493525 550011 0 0 0 2260137 11 7 1 13860 103950 39963 0 0 0 157773 11 1 8 4950 13200 0 0 0 0 18150 11 2 7 105435 259017 0 0 0 0 364452 11 3 6 634865 1493525 0 0 0 0 2128390 11 4 5 1482250 3420835 0 0 0 0 4903085 11 5 4 1482250 3420835 0 0 0 0 4903085 11 6 3 634865 1493525 0 0 0 0 2128390 11 7 2 105435 259017 0 0 0 0 364452 11 8 1 4950 13200 0 0 0 0 18150 11 1 9 825 495 0 0 0 0 1320 11 2 8 23694 13200 0 0 0 0 36894 11 3 7 194304 103950 0 0 0 0 298254 11 4 6 634865 332640 0 0 0 0 967505 11 5 5 931854 485100 0 0 0 0 1416954 11 6 4 634865 332640 0 0 0 0 967505 11 7 3 194304 103950 0 0 0 0 298254 11 8 2 23694 13200 0 0 0 0 36894 11 9 1 825 495 0 0 0 0 1320 11 1 10 55 0 0 0 0 0 55 11 2 9 2145 0 0 0 0 0 2145 11 3 8 23694 0 0 0 0 0 23694 11 4 7 105435 0 0 0 0 0 105435 11 5 6 216601 0 0 0 0 0 216601 11 6 5 216601 0 0 0 0 0 216601 11 7 4 105435 0 0 0 0 0 105435 11 8 3 23694 0 0 0 0 0 23694 11 9 2 2145 0 0 0 0 0 2145 11 10 1 55 0 0 0 0 0 55 11 1 11 1 0 0 0 0 0 1 11 2 10 55 0 0 0 0 0 55 11 3 9 825 0 0 0 0 0 825 11 4 8 4950 0 0 0 0 0 4950 11 5 7 13860 0 0 0 0 0 13860 11 6 6 19404 0 0 0 0 0 19404 11 7 5 13860 0 0 0 0 0 13860 11 8 4 4950 0 0 0 0 0 4950 11 9 3 825 0 0 0 0 0 825 11 10 2 55 0 0 0 0 0 55 11 11 1 1 0 0 0 0 0 1 11 sum 13891584 81968469 160174960 112868844 23235300 604800 392743957 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 g=5 all g 11 1 1 1 45 3633 63355 207876 54990 329900 11 1 2 5 1200 50001 418810 496224 0 966240 11 2 1 5 1200 50001 418810 496224 0 966240 11 1 3 75 9450 201915 740100 207876 0 1159416 11 2 2 195 23547 491729 1779193 496224 0 2790888 11 3 1 75 9450 201915 740100 207876 0 1159416 11 1 4 450 30240 315000 418810 0 0 764500 11 2 3 2154 135775 1364986 1779193 0 0 3282108 11 3 2 2154 135775 1364986 1779193 0 0 3282108 11 4 1 450 30240 315000 418810 0 0 764500 11 1 5 1260 44100 201915 63355 0 0 310630 11 2 4 9585 310985 1364986 418810 0 0 2104366 11 3 3 17664 560498 2428862 740100 0 0 3747124 11 4 2 9585 310985 1364986 418810 0 0 2104366 11 5 1 1260 44100 201915 63355 0 0 310630 11 1 6 1764 30240 50001 0 0 0 82005 11 2 5 19691 310985 491729 0 0 0 822405 11 3 4 57715 880403 1364986 0 0 0 2303104 11 4 3 57715 880403 1364986 0 0 0 2303104 11 5 2 19691 310985 491729 0 0 0 822405 11 6 1 1764 30240 50001 0 0 0 82005 11 1 7 1260 9450 3633 0 0 0 14343 11 2 6 19691 135775 50001 0 0 0 205467 11 3 5 84714 560498 201915 0 0 0 847127 11 4 4 134750 880403 315000 0 0 0 1330153 11 5 3 84714 560498 201915 0 0 0 847127 11 6 2 19691 135775 50001 0 0 0 205467 11 7 1 1260 9450 3633 0 0 0 14343 11 1 8 450 1200 0 0 0 0 1650 11 2 7 9585 23547 0 0 0 0 33132 11 3 6 57715 135775 0 0 0 0 193490 11 4 5 134750 310985 0 0 0 0 445735 11 5 4 134750 310985 0 0 0 0 445735 11 6 3 57715 135775 0 0 0 0 193490 11 7 2 9585 23547 0 0 0 0 33132 11 8 1 450 1200 0 0 0 0 1650 11 1 9 75 45 0 0 0 0 120 11 2 8 2154 1200 0 0 0 0 3354 11 3 7 17664 9450 0 0 0 0 27114 11 4 6 57715 30240 0 0 0 0 87955 11 5 5 84714 44100 0 0 0 0 128814 11 6 4 57715 30240 0 0 0 0 87955 11 7 3 17664 9450 0 0 0 0 27114 11 8 2 2154 1200 0 0 0 0 3354 11 9 1 75 45 0 0 0 0 120 11 1 10 5 0 0 0 0 0 5 11 2 9 195 0 0 0 0 0 195 11 3 8 2154 0 0 0 0 0 2154 11 4 7 9585 0 0 0 0 0 9585 11 5 6 19691 0 0 0 0 0 19691 11 6 5 19691 0 0 0 0 0 19691 11 7 4 9585 0 0 0 0 0 9585 11 8 3 2154 0 0 0 0 0 2154 11 9 2 195 0 0 0 0 0 195 11 10 1 5 0 0 0 0 0 5 11 1 11 1 0 0 0 0 0 1 11 2 10 5 0 0 0 0 0 5 11 3 9 75 0 0 0 0 0 75 11 4 8 450 0 0 0 0 0 450 11 5 7 1260 0 0 0 0 0 1260 11 6 6 1764 0 0 0 0 0 1764 11 7 5 1260 0 0 0 0 0 1260 11 8 4 450 0 0 0 0 0 450 11 9 3 75 0 0 0 0 0 75 11 10 2 5 0 0 0 0 0 5 11 11 1 1 0 0 0 0 0 1 11 sum 1262874 7451679 14561360 10260804 2112300 54990 35704007 Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 g=5 all g 11 1 1 1 30 1894 32028 104748 28169 166870 11 1 2 5 650 25442 211149 250674 0 487920 11 2 1 5 650 25442 211149 250674 0 487920 11 1 3 50 4890 102033 372579 104748 0 584300 11 2 2 145 12507 250375 899919 250674 0 1413620 11 3 1 50 4890 102033 372579 104748 0 584300 11 1 4 250 15429 158902 211149 0 0 385730 11 2 3 1272 70364 692895 899919 0 0 1664450 11 3 2 1272 70364 692895 899919 0 0 1664450 11 4 1 250 15429 158902 211149 0 0 385730 11 1 5 680 22439 102033 32028 0 0 157180 11 2 4 5280 159881 692895 211149 0 0 1069205 11 3 3 9895 289690 1235766 372579 0 0 1907930 11 4 2 5280 159881 692895 211149 0 0 1069205 11 5 1 680 22439 102033 32028 0 0 157180 11 1 6 932 15429 25442 0 0 0 41803 11 2 5 10580 159881 250375 0 0 0 420836 11 3 4 31276 453914 692895 0 0 0 1178085 11 4 3 31276 453914 692895 0 0 0 1178085 11 5 2 10580 159881 250375 0 0 0 420836 11 6 1 932 15429 25442 0 0 0 41803 11 1 7 680 4890 1894 0 0 0 7464 11 2 6 10580 70364 25442 0 0 0 106386 11 3 5 45593 289690 102033 0 0 0 437316 11 4 4 72417 453914 158902 0 0 0 685233 11 5 3 45593 289690 102033 0 0 0 437316 11 6 2 10580 70364 25442 0 0 0 106386 11 7 1 680 4890 1894 0 0 0 7464 11 1 8 250 650 0 0 0 0 900 11 2 7 5280 12507 0 0 0 0 17787 11 3 6 31276 70364 0 0 0 0 101640 11 4 5 72417 159881 0 0 0 0 232298 11 5 4 72417 159881 0 0 0 0 232298 11 6 3 31276 70364 0 0 0 0 101640 11 7 2 5280 12507 0 0 0 0 17787 11 8 1 250 650 0 0 0 0 900 11 1 9 50 30 0 0 0 0 80 11 2 8 1272 650 0 0 0 0 1922 11 3 7 9895 4890 0 0 0 0 14785 11 4 6 31276 15429 0 0 0 0 46705 11 5 5 45593 22439 0 0 0 0 68032 11 6 4 31276 15429 0 0 0 0 46705 11 7 3 9895 4890 0 0 0 0 14785 11 8 2 1272 650 0 0 0 0 1922 11 9 1 50 30 0 0 0 0 80 11 1 10 5 0 0 0 0 0 5 11 2 9 145 0 0 0 0 0 145 11 3 8 1272 0 0 0 0 0 1272 11 4 7 5280 0 0 0 0 0 5280 11 5 6 10580 0 0 0 0 0 10580 11 6 5 10580 0 0 0 0 0 10580 11 7 4 5280 0 0 0 0 0 5280 11 8 3 1272 0 0 0 0 0 1272 11 9 2 145 0 0 0 0 0 145 11 10 1 5 0 0 0 0 0 5 11 1 11 1 0 0 0 0 0 1 11 2 10 5 0 0 0 0 0 5 11 3 9 50 0 0 0 0 0 50 11 4 8 250 0 0 0 0 0 250 11 5 7 680 0 0 0 0 0 680 11 6 6 932 0 0 0 0 0 932 11 7 5 680 0 0 0 0 0 680 11 8 4 250 0 0 0 0 0 250 11 9 3 50 0 0 0 0 0 50 11 10 2 5 0 0 0 0 0 5 11 11 1 1 0 0 0 0 0 1 11 sum 683307 3843024 7391499 5180472 1066266 28169 18192737 Number of rooted hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 g=5 all g 12 1 1 1 715 88803 2641925 18128396 19056960 39916800 12 1 2 66 23595 1585584 24656775 75220860 19056960 120543840 12 2 1 66 23595 1585584 24656775 75220860 19056960 120543840 12 1 3 1210 235950 8654646 66805310 75220860 0 150917976 12 2 2 3135 585585 20981337 159762815 178462816 0 359795688 12 3 1 1210 235950 8654646 66805310 75220860 0 150917976 12 1 4 9075 990990 19324305 66805310 18128396 0 105258076 12 2 3 43098 4410120 82897296 280514670 75220860 0 443086044 12 3 2 43098 4410120 82897296 280514670 75220860 0 443086044 12 4 1 9075 990990 19324305 66805310 18128396 0 105258076 12 1 5 32670 1981980 19324305 24656775 0 0 45995730 12 2 4 245223 13768300 128420004 159762815 0 0 302196342 12 3 3 449988 24695580 227256510 280514670 0 0 532916748 12 4 2 245223 13768300 128420004 159762815 0 0 302196342 12 5 1 32670 1981980 19324305 24656775 0 0 45995730 12 1 6 60984 1981980 8654646 2641925 0 0 13339535 12 2 5 666996 19920390 82897296 24656775 0 0 128141457 12 3 4 1936308 55785870 227256510 66805310 0 0 351783998 12 4 3 1936308 55785870 227256510 66805310 0 0 351783998 12 5 2 666996 19920390 82897296 24656775 0 0 128141457 12 6 1 60984 1981980 8654646 2641925 0 0 13339535 12 1 7 60984 990990 1585584 0 0 0 2637558 12 2 6 925190 13768300 20981337 0 0 0 35674827 12 3 5 3915576 55785870 82897296 0 0 0 142598742 12 4 4 6195560 87100531 128420004 0 0 0 221716095 12 5 3 3915576 55785870 82897296 0 0 0 142598742 12 6 2 925190 13768300 20981337 0 0 0 35674827 12 7 1 60984 990990 1585584 0 0 0 2637558 12 1 8 32670 235950 88803 0 0 0 357423 12 2 7 666996 4410120 1585584 0 0 0 6662700 12 3 6 3915576 24695580 8654646 0 0 0 37265802 12 4 5 9032898 55785870 19324305 0 0 0 84143073 12 5 4 9032898 55785870 19324305 0 0 0 84143073 12 6 3 3915576 24695580 8654646 0 0 0 37265802 12 7 2 666996 4410120 1585584 0 0 0 6662700 12 8 1 32670 235950 88803 0 0 0 357423 12 1 9 9075 23595 0 0 0 0 32670 12 2 8 245223 585585 0 0 0 0 830808 12 3 7 1936308 4410120 0 0 0 0 6346428 12 4 6 6195560 13768300 0 0 0 0 19963860 12 5 5 9032898 19920390 0 0 0 0 28953288 12 6 4 6195560 13768300 0 0 0 0 19963860 12 7 3 1936308 4410120 0 0 0 0 6346428 12 8 2 245223 585585 0 0 0 0 830808 12 9 1 9075 23595 0 0 0 0 32670 12 1 10 1210 715 0 0 0 0 1925 12 2 9 43098 23595 0 0 0 0 66693 12 3 8 449988 235950 0 0 0 0 685938 12 4 7 1936308 990990 0 0 0 0 2927298 12 5 6 3915576 1981980 0 0 0 0 5897556 12 6 5 3915576 1981980 0 0 0 0 5897556 12 7 4 1936308 990990 0 0 0 0 2927298 12 8 3 449988 235950 0 0 0 0 685938 12 9 2 43098 23595 0 0 0 0 66693 12 10 1 1210 715 0 0 0 0 1925 12 1 11 66 0 0 0 0 0 66 12 2 10 3135 0 0 0 0 0 3135 12 3 9 43098 0 0 0 0 0 43098 12 4 8 245223 0 0 0 0 0 245223 12 5 7 666996 0 0 0 0 0 666996 12 6 6 925190 0 0 0 0 0 925190 12 7 5 666996 0 0 0 0 0 666996 12 8 4 245223 0 0 0 0 0 245223 12 9 3 43098 0 0 0 0 0 43098 12 10 2 3135 0 0 0 0 0 3135 12 11 1 66 0 0 0 0 0 66 12 1 12 1 0 0 0 0 0 1 12 2 11 66 0 0 0 0 0 66 12 3 10 1210 0 0 0 0 0 1210 12 4 9 9075 0 0 0 0 0 9075 12 5 8 32670 0 0 0 0 0 32670 12 6 7 60984 0 0 0 0 0 60984 12 7 6 60984 0 0 0 0 0 60984 12 8 5 32670 0 0 0 0 0 32670 12 9 4 9075 0 0 0 0 0 9075 12 10 3 1210 0 0 0 0 0 1210 12 11 2 66 0 0 0 0 0 66 12 12 1 1 0 0 0 0 0 1 12 sum 91287552 685888171 1805010948 1877530740 684173164 57170880 5201061455 Number of sensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 g=5 all g 12 1 1 1 62 7417 220244 1510846 1588218 3326788 12 1 2 6 1976 132202 2054974 6268712 1588218 10046088 12 2 1 6 1976 132202 2054974 6268712 1588218 10046088 12 1 3 104 19694 721382 5567550 6268712 0 12577442 12 2 2 265 48846 1748723 13314231 14872428 0 29984493 12 3 1 104 19694 721382 5567550 6268712 0 12577442 12 1 4 765 82652 1610617 5567550 1510846 0 8772430 12 2 3 3605 367645 6908644 23377106 6268712 0 36925712 12 3 2 3605 367645 6908644 23377106 6268712 0 36925712 12 4 1 765 82652 1610617 5567550 1510846 0 8772430 12 1 5 2736 165262 1610617 2054974 0 0 3833589 12 2 4 20472 1147628 10702449 13314231 0 0 25184780 12 3 3 37545 2058329 18938994 23377106 0 0 44411974 12 4 2 20472 1147628 10702449 13314231 0 0 25184780 12 5 1 2736 165262 1610617 2054974 0 0 3833589 12 1 6 5102 165262 721382 220244 0 0 1111990 12 2 5 55633 1660331 6908644 2054974 0 0 10679582 12 3 4 161455 4649379 18938994 5567550 0 0 29317378 12 4 3 161455 4649379 18938994 5567550 0 0 29317378 12 5 2 55633 1660331 6908644 2054974 0 0 10679582 12 6 1 5102 165262 721382 220244 0 0 1111990 12 1 7 5102 82652 132202 0 0 0 219956 12 2 6 77174 1147628 1748723 0 0 0 2973525 12 3 5 326432 4649379 6908644 0 0 0 11884455 12 4 4 516507 7259140 10702449 0 0 0 18478096 12 5 3 326432 4649379 6908644 0 0 0 11884455 12 6 2 77174 1147628 1748723 0 0 0 2973525 12 7 1 5102 82652 132202 0 0 0 219956 12 1 8 2736 19694 7417 0 0 0 29847 12 2 7 55633 367645 132202 0 0 0 555480 12 3 6 326432 2058329 721382 0 0 0 3106143 12 4 5 752940 4649379 1610617 0 0 0 7012936 12 5 4 752940 4649379 1610617 0 0 0 7012936 12 6 3 326432 2058329 721382 0 0 0 3106143 12 7 2 55633 367645 132202 0 0 0 555480 12 8 1 2736 19694 7417 0 0 0 29847 12 1 9 765 1976 0 0 0 0 2741 12 2 8 20472 48846 0 0 0 0 69318 12 3 7 161455 367645 0 0 0 0 529100 12 4 6 516507 1147628 0 0 0 0 1664135 12 5 5 752940 1660331 0 0 0 0 2413271 12 6 4 516507 1147628 0 0 0 0 1664135 12 7 3 161455 367645 0 0 0 0 529100 12 8 2 20472 48846 0 0 0 0 69318 12 9 1 765 1976 0 0 0 0 2741 12 1 10 104 62 0 0 0 0 166 12 2 9 3605 1976 0 0 0 0 5581 12 3 8 37545 19694 0 0 0 0 57239 12 4 7 161455 82652 0 0 0 0 244107 12 5 6 326432 165262 0 0 0 0 491694 12 6 5 326432 165262 0 0 0 0 491694 12 7 4 161455 82652 0 0 0 0 244107 12 8 3 37545 19694 0 0 0 0 57239 12 9 2 3605 1976 0 0 0 0 5581 12 10 1 104 62 0 0 0 0 166 12 1 11 6 0 0 0 0 0 6 12 2 10 265 0 0 0 0 0 265 12 3 9 3605 0 0 0 0 0 3605 12 4 8 20472 0 0 0 0 0 20472 12 5 7 55633 0 0 0 0 0 55633 12 6 6 77174 0 0 0 0 0 77174 12 7 5 55633 0 0 0 0 0 55633 12 8 4 20472 0 0 0 0 0 20472 12 9 3 3605 0 0 0 0 0 3605 12 10 2 265 0 0 0 0 0 265 12 11 1 6 0 0 0 0 0 6 12 1 12 1 0 0 0 0 0 1 12 2 11 6 0 0 0 0 0 6 12 3 10 104 0 0 0 0 0 104 12 4 9 765 0 0 0 0 0 765 12 5 8 2736 0 0 0 0 0 2736 12 6 7 5102 0 0 0 0 0 5102 12 7 6 5102 0 0 0 0 0 5102 12 8 5 2736 0 0 0 0 0 2736 12 9 4 765 0 0 0 0 0 765 12 10 3 104 0 0 0 0 0 104 12 11 2 6 0 0 0 0 0 6 12 12 1 1 0 0 0 0 0 1 12 sum 7611156 57167260 150429819 156469887 57017238 4764654 433460014 \end{verbatim} \end{scriptsize} \newpage \begin{scriptsize} \begin{verbatim} Number of unsensed hypermaps with d darts, v vertices and e edges. d v e g=0 g=1 g=2 g=3 g=4 g=5 all g 12 1 1 1 41 3836 110914 757977 797345 1670114 12 1 2 6 1058 66865 1031387 3142703 797345 5039364 12 2 1 6 1058 66865 1031387 3142703 797345 5039364 12 1 3 67 10107 362868 2791448 3142703 0 6307193 12 2 2 195 25594 883711 6689591 7472556 0 15071647 12 3 1 67 10107 362868 2791448 3142703 0 6307193 12 1 4 420 41890 808812 2791448 757977 0 4400547 12 2 3 2086 188410 3481842 11744994 3142703 0 18560035 12 3 2 2086 188410 3481842 11744994 3142703 0 18560035 12 4 1 420 41890 808812 2791448 757977 0 4400547 12 1 5 1443 83460 808812 1031387 0 0 1925102 12 2 4 11060 583755 5389906 6689591 0 0 12674312 12 3 3 20565 1050920 9551009 11744994 0 0 22367488 12 4 2 11060 583755 5389906 6689591 0 0 12674312 12 5 1 1443 83460 808812 1031387 0 0 1925102 12 1 6 2651 83460 362868 110914 0 0 559893 12 2 5 29237 842635 3481842 1031387 0 0 5385101 12 3 4 85673 2366909 9551009 2791448 0 0 14795039 12 4 3 85673 2366909 9551009 2791448 0 0 14795039 12 5 2 29237 842635 3481842 1031387 0 0 5385101 12 6 1 2651 83460 362868 110914 0 0 559893 12 1 7 2651 41890 66865 0 0 0 111406 12 2 6 40348 583755 883711 0 0 0 1507814 12 3 5 171275 2366909 3481842 0 0 0 6020026 12 4 4 271482 3696390 5389906 0 0 0 9357778 12 5 3 171275 2366909 3481842 0 0 0 6020026 12 6 2 40348 583755 883711 0 0 0 1507814 12 7 1 2651 41890 66865 0 0 0 111406 12 1 8 1443 10107 3836 0 0 0 15386 12 2 7 29237 188410 66865 0 0 0 284512 12 3 6 171275 1050920 362868 0 0 0 1585063 12 4 5 394258 2366909 808812 0 0 0 3569979 12 5 4 394258 2366909 808812 0 0 0 3569979 12 6 3 171275 1050920 362868 0 0 0 1585063 12 7 2 29237 188410 66865 0 0 0 284512 12 8 1 1443 10107 3836 0 0 0 15386 12 1 9 420 1058 0 0 0 0 1478 12 2 8 11060 25594 0 0 0 0 36654 12 3 7 85673 188410 0 0 0 0 274083 12 4 6 271482 583755 0 0 0 0 855237 12 5 5 394258 842635 0 0 0 0 1236893 12 6 4 271482 583755 0 0 0 0 855237 12 7 3 85673 188410 0 0 0 0 274083 12 8 2 11060 25594 0 0 0 0 36654 12 9 1 420 1058 0 0 0 0 1478 12 1 10 67 41 0 0 0 0 108 12 2 9 2086 1058 0 0 0 0 3144 12 3 8 20565 10107 0 0 0 0 30672 12 4 7 85673 41890 0 0 0 0 127563 12 5 6 171275 83460 0 0 0 0 254735 12 6 5 171275 83460 0 0 0 0 254735 12 7 4 85673 41890 0 0 0 0 127563 12 8 3 20565 10107 0 0 0 0 30672 12 9 2 2086 1058 0 0 0 0 3144 12 10 1 67 41 0 0 0 0 108 12 1 11 6 0 0 0 0 0 6 12 2 10 195 0 0 0 0 0 195 12 3 9 2086 0 0 0 0 0 2086 12 4 8 11060 0 0 0 0 0 11060 12 5 7 29237 0 0 0 0 0 29237 12 6 6 40348 0 0 0 0 0 40348 12 7 5 29237 0 0 0 0 0 29237 12 8 4 11060 0 0 0 0 0 11060 12 9 3 2086 0 0 0 0 0 2086 12 10 2 195 0 0 0 0 0 195 12 11 1 6 0 0 0 0 0 6 12 1 12 1 0 0 0 0 0 1 12 2 11 6 0 0 0 0 0 6 12 3 10 67 0 0 0 0 0 67 12 4 9 420 0 0 0 0 0 420 12 5 8 1443 0 0 0 0 0 1443 12 6 7 2651 0 0 0 0 0 2651 12 7 6 2651 0 0 0 0 0 2651 12 8 5 1443 0 0 0 0 0 1443 12 9 4 420 0 0 0 0 0 420 12 10 3 67 0 0 0 0 0 67 12 11 2 6 0 0 0 0 0 6 12 12 1 1 0 0 0 0 0 1 12 sum 4004055 29107494 75807708 78573507 28602705 2392035 218487504 \end{verbatim} \end{scriptsize} \bibliographystyle{jis} \begin{thebibliography}{10} \bibitem{Arq86a} D.~Arqu{\`e}s, Relations fonctionnelles et d\'enombrement des hypercartes planaires point\'ees, in G.~Labelle and P.~Leroux, eds., {\em Combinatoire \'Enum\'erative}, Lecture Notes in Math., Vol.~1234, Springer, 1986, pp.~5--26. \bibitem{Arq87} D.~Arqu{\`e}s, Hypercartes point\'ees sur le tore : d\'ecompositions et d\'enombrements, {\em J. Combin. Theory Ser. B} {\bf 43} (1987), 275--286. \bibitem{Arq87a} D.~Arqu{\`e}s, Relations fonctionnelles et d{\'e}nombrement des cartes point{\'e}es sur le tore, {\em J. Combin. Theory Ser. B} {\bf 43} (1987), 253--274. \bibitem{BC86} E.~A. Bender and E.~R. Canfield, The asymptotic number of rooted maps on a surface, {\em J. Combin. Theory Ser. A} {\bf 43} (1986), 244--257. \bibitem{BC91} E.~A. Bender and E.~R. Canfield, The number of rooted maps on an orientable surface, {\em J. Combin. Theory Ser. B} {\bf 53} (1991), 293--299. \bibitem{CC14-v3} S.~R. Carrell and G.~Chapuy, Simple recurrence formulas to count maps on orientable surfaces. Preprint, \url{http://arxiv.org/abs/1402.6300}, 2014. \bibitem{Chauve03} C.~Chauve, A bijection between planar constellations and some colored {L}agrangian trees, {\em Discrete Math. Theor. Comput. Sci.} {\bf 6} (2003), 13--40. \bibitem{Cox73} H.~S.~M. Coxeter, {\em {Regular Polytopes}}, 3rd ed., Dover, 1973. \bibitem{gio98a} A.~Giorgetti, {\em Combinatoire bijective et \'enum\'erative des cartes point\'ees sur une surface}, Ph.~D. thesis, Universit\'e de Marne-la-Vall\'ee, Institut Gaspard Monge, 1998. \newblock \url{http://tel.archives-ouvertes.fr/tel-00724977}. \bibitem{GW14} A.~Giorgetti and T.~R.~S. Walsh, Constructing large tables of numbers of maps by orientable genus, preprint, \url{http://arxiv.org/abs/1405.0615}, 2014. \bibitem{GJ83} I.~P. Goulden and D.~M. Jackson, {\em Combinatorial Enumeration}, Wiley, 1983. \bibitem{GLM97} T.~Gr\"uner, L.~Laue, and M.~Meringer, Algorithms for group actions: homomorphism principle and orderly generation applied to graphs, {\em DIMACS Ser. Discrete Math. Theoret. Comput. Sci.} {\bf 28} (1997), 113--122. \bibitem{JV01} D.~M. Jackson and T.~I. Visentin, {\em {An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces}}, Chapman and Hall/CRC, 2001. \bibitem{JoSi} G.~A. Jones and D.~Singerman, Theory of maps on orientable surfaces, {\em Proc. Lond. Math. Soc.} {\bf 37} (1978), 273--307. \bibitem{LZ04} S.~K. Lando and A.~K. Zvonkin, {\em {Graphs on Surfaces and Their Applications}}, Springer, 2004. \bibitem{Liskovets81} V.~A. Liskovets, A census of nonisomorphic planar maps, in L.~Lov\'asz and V.~T. S\'os, eds., {\em Algebraic Methods in Graph Theory}, Vol.~2, North-Holland, 1981, pp.~479--494. \bibitem{MG11} A.~Mednykh and A.~Giorgetti, Enumeration of genus four maps by number of edges, {\em Ars Math. Contemp.} {\bf 4} (2011), 351--361. \bibitem{MN06} A.~Mednykh and R.~Nedela, Enumeration of unrooted maps of a given genus, {\em J. Combin. Theory Ser. B} {\bf 96} (2006), 706--729. \bibitem{MN10} A.~Mednykh and R.~Nedela, Enumeration of unrooted hypermaps of a given genus, {\em Discrete Math.} {\bf 310} (2010), 518--526. \bibitem{Read78} R.~C. Read, Every one a winner, {\em Annals of Discrete Math.} {\bf 2} (1978), 107--120. \bibitem{RW95} L.~B. Richmond and N.~C. Wormald, Almost all maps are asymmetric, {\em J. Combin. Theory Ser. {B}} {\bf 63} (1995), 1--7. \bibitem{Tarjan72} R.~E. Tarjan, Depth-first search and linear graph algorithms, {\em {SIAM} J. Comput.} {\bf 1} (1972), 146--160. \bibitem{Tarry95} G.~Tarry, Le probl\`eme des labyrinthes, {\em Nouvelles Annales de Math\'ematiques} {\bf 3} (1895), 187--‑190. \bibitem{Tu1} W.~T. Tutte, A census of planar maps, {\em Canad. J. Math.} {\bf 15} (1963), 249--271. \bibitem{Tu3} W.~T. Tutte, On the enumeration of planar maps, {\em Bull. Amer. Math. Soc.} {\bf 74} (1968), 64--74. \bibitem{WalshCode} T.~R.~S. Walsh, {Source code for the C program that generates nonisomorphic maps and hypermaps}. \newline \url{http://www.info2.uqam.ca/~walsh_t/programs/map generating program.txt}. \bibitem{Walsh71} T.~R.~S. Walsh, {\em Combinatorial enumeration of non-planar maps}, Ph.~D. thesis, University of Toronto, 1971. \bibitem{Walsh75} T.~R.~S. Walsh, Hypermaps vs. bipartite maps, {\em J. Combin. Theory Ser. B} {\bf 18} (1975), 155--163. \bibitem{Walsh83b} T.~R.~S. Walsh, Generating nonisomorphic maps without storing them, {\em SIAM J. Alg. Disc. Meth.} {\bf 4} (1983), 161--178. \bibitem{Walsh83} T.~R.~S. Walsh, A polynomial algorithm for counting rooted toroidal maps, {\em Ars Combin.} {\bf 16} (1983), 49--56. \bibitem{Walsh13} T.~R.~S. Walsh, {Counting maps on doughnuts, invited talk at the 2010 GASCom conference at UQAM}, {\em Theoret. Comput. Sci.} {\bf 502} (2013), 4--15. \bibitem{WG14} T.~R.~S. Walsh and A.~Giorgetti, Efficient enumeration of rooted maps of a given orientable genus by number of faces and vertices, {\em Ars Math. Contemp.} {\bf 7} (2014), 263--280. \bibitem{WGMunrooted} T.~R.~S. Walsh, A.~Giorgetti, and A.~Mednykh, Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices, {\em Discrete Math.} {\bf 312} (2012), 2660--2671. \bibitem{W1} T.~R.~S. Walsh and A.~B. Lehman, Counting rooted maps by genus {I}, {\em J. Combin. Theory Ser. B} {\bf 13} (1972), 192--218. \bibitem{Wormald81b} N.~C. Wormald, Counting unrooted planar maps, {\em Discrete Math.} {\bf 36} (1981), 205--225. \bibitem{Wormald81a} N.~C. Wormald, On the number of planar maps, {\em Canad. J. Math.} {\bf 33} (1981), 1--11. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 05C30; Secondary 05C10, 05C65, 05C85. \noindent \emph{Keywords: } map, hypermap, exhaustive generation, isomorphism class. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A006385}, \seqnum{A006387}, \seqnum{A214814}, \seqnum{A214815}, \seqnum{A214816}, \seqnum{A000257}, \seqnum{A118093}, \seqnum{A214817}, \seqnum{A214818}, \seqnum{A003319}, \seqnum{A090371}, \seqnum{A118094}, \seqnum{A214819}, \seqnum{A214820}, \seqnum{A057005}, \seqnum{A214821}, \seqnum{A214822}, \seqnum{A214823}, \seqnum{A215017}, and \seqnum{A215018}.) \vspace*{+.1in} \noindent Received October 3 2014; revised versions received October 6 2014; October 29 2014; February 19 2015; February 23 2015. Published in {\it Journal of Integer Sequences}, May 12 2015. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}