\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} \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}}} \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} \newcommand{\hz}{\widehat{0}} \newcommand{\ho}{\widehat{1}} \newcommand{\timesdots}{\times \cdots \times} \newcommand{\coveredby}{\prec} \DeclareMathOperator{\inv}{inv} \DeclareMathOperator{\frst}{frst} \DeclareMathOperator{\er}{er} \DeclareMathOperator{\asc}{asc} \DeclareMathOperator{\odd}{odd} \newcommand{\ascodd}{\asc_{\odd}} \newcommand{\doubleprime}{\prime\prime} \newcommand{\SSSS}{\mathfrak{S}} \newcommand{\Gaussian}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}_q} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf The Gaussian Coefficient Revisited\\ \vskip .1in } \vskip 1cm \large Richard Ehrenborg and Margaret A.\ Readdy \\ Department of Mathematics \\ University of Kentucky \\ Lexington, KY 40506-0027 \\ USA \\ \href{mailto:richard.ehrenborg@uky.edu}{\tt richard.ehrenborg@uky.edu} \\ \href{mailto:margaret.readdy@uky.edu}{\tt margaret.readdy@uky.edu} \\ \end{center} \vskip .2 in \begin{abstract} We give a new $q$-$(1+q)$-analogue of the Gaussian coefficient, also known as the $q$-binomial which, like the original $q$-binomial $\Gaussian{n}{k}$, is symmetric in $k$ and $n-k$. We show this $q$-$(1+q)$-binomial is more compact than the one discovered by Fu, Reiner, Stanton, and Thiem. Underlying our $q$-$(1+q)$-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birkhoff transform of any finite poset. We end with a discussion of higher analogues of the $q$-binomial. \end{abstract} \section{Introduction} Inspired by work of Fu, Reiner, Stanton, and Thiem~\cite{Fu_Reiner_Stanton_Thiem}, Cai and Readdy~\cite{Cai_Readdy} asked the following question. Given a combinatorial $q$-analogue $$ X(q) = \sum_{w \in X} q^{a(w)} , $$ where $X$ is a set of objects and $a(\cdot)$ is a statistic defined on the elements of $X$, when can one find a smaller set $Y$ and two statistics $s$ and $t$ such that $$ X(q) = \sum_{w \in Y} q^{s(w)} \cdot (1+q)^{t(w)} . $$ Such an interpretation is called an {\em $q$-$(1+q)$-analogue}. Examples of $q$-$(1+q)$-analogues have been determined for the $q$-binomial by Fu et al.~\cite{Fu_Reiner_Stanton_Thiem}, and for the $q$-Stirling numbers of the first and second kinds by Cai and Readdy~\cite{Cai_Readdy}, who also gave poset and homotopy interpretations of their $q$-$(1+q)$-analogues. In 1916 MacMahon~\cite{MacMahon_1,MacMahon_collected,MacMahon_book} observed that the Gaussian coefficient, also known as the $q$-binomial coefficient, is given by $$ \Gaussian{n}{k} = \sum_{w \in \Omega_{n,k}} q^{\inv(w)} . $$ Here $\Omega_{n,k} = \SSSS(0^{n-k},1^{k})$ denotes all permutations of the multiset $\{0^{n-k},1^{k}\}$, that is, all words $w = w_1 \cdots w_n$ of length~$n$ with $n-k$ zeroes and $k$ ones, and $\inv( \cdot )$ denotes the inversion statistic defined by $\inv(w_{1} w_{2} \cdots w_{n}) = |\{(i,j) \: : \: 1 \leq i < j \leq n, w_{i} > w_{j}\}|$. Fu et al.\ defined a subset $\Omega^{\prime}_{n,k} \subseteq \Omega_{n,k}$ and two statistics $a$ and $b$ such that $$ \Gaussian{n}{k} = \sum_{w \in \Omega^{\prime}_{n,k}} q^{a(w)} \cdot (1+q)^{b(w)} . $$ In this paper we will return to the original study by Fu et al.\ of the Gaussian coefficient. We discover a more compact $q$-$(1+q)$-analogue which, like the original Gaussian coefficient, is also symmetric in the variables $k$ and $n-k$. See Corollary~\ref{corollary_symmetry} and Theorem~\ref{theorem_compact}. This symmetry was missing in Fu et al.'s original $q$-$(1+q)$-analogue. We give a Boolean algebra decomposition of the related poset $\Omega_{n,k}$. Since this poset is a distributive lattice, in the last section we extend these ideas to poset decompositions of any distributive lattice and other analogues. \section{A poset interpretation} In this section we consider the poset structure on $0$-$1$-words in $\Omega_{n,k}$. For further poset terminology and background, we refer the reader to~\cite{EC1}. We begin by making the set of elements $\Omega_{n,k}$ into a graded poset by defining the cover relation to be $$ u \circ 01 \circ v \prec u \circ 10 \circ v , $$ where $\circ$ denotes concatenation of words. The word $0^{n-k} 1^{k}$ is the minimal element and the word $1^{k} 0^{n-k}$ is the maximal element in the poset $\Omega_{n,k}$. Furthermore, this poset is graded by the inversion statistic. This poset is simply the interval $[\hz, x]$ of Young's lattice, where the minimal element $\hz$ is the empty Ferrers diagram and $x$ is the Ferrers diagram consisting of $n-k$ columns and $k$ rows. An alternative description of the poset $\Omega_{n,k}$ is that it is isomorphic to the Birkhoff transform of the Cartesian product of two chains. Let $C_{m}$ denote the $m$-element chain. The poset $\Omega_{n,k}$ is isomorphic to the distributive lattice of all lower order ideals of the product $C_{n-k} \times C_{k}$, usually denoted by $J(C_{n-k} \times C_{k})$. \begin{definition} Let $\Omega^{\doubleprime}_{n,k}$ consist of all $0$,$1$-words $v = v_{1} v_{2} \cdots v_{n}$ in $\Omega_{n,k}$ such that $$ v_{1} \leq v_{2}, \:\: v_{3} \leq v_{4}, \:\: \ldots, \:\: v_{2 \cdot \lfloor n/2 \rfloor - 1} \leq v_{2 \cdot \lfloor n/2 \rfloor} . $$ \end{definition} Observe that when $n$ is odd there is no condition on the last entry $w_{n}$. Define two maps $\phi$ and~$\psi$ on $\Omega_{n,k}$ by sending the word $w = w_{1} w_{2} \cdots w_{n}$ to \begin{align*} \phi(w) &= \min(w_{1},w_{2}), \max(w_{1},w_{2}), \:\: \min(w_{3},w_{4}), \max(w_{3},w_{4}), \:\: \ldots , \\ \psi(w) &= \max(w_{1},w_{2}), \min(w_{1},w_{2}), \:\: \max(w_{3},w_{4}), \min(w_{3},w_{4}), \:\: \ldots . \end{align*} The map $\phi$ sorts the entries in positions $1$ and $2$, $3$ and $4$, and so on. If $n$ is odd, the entry~$w_{n}$ remains in the same position. Similarly, the map $\psi$ sorts in reverse order each pair of positions. Note that the map $\phi$ maps $\Omega_{n,k}$ surjectively onto the set $\Omega^{\doubleprime}_{n,k}$. We have the following Boolean algebra decomposition of the poset $\Omega_{n,k}$. \begin{theorem} The distributive lattice $\Omega_{n,k}$ has the Boolean algebra decomposition $$ \Omega_{n,k} = \bigcup_{v \in \Omega^{\doubleprime}_{n,k}} [v,\psi(v)] . $$ \label{theorem_decomposition} \end{theorem} \begin{proof} Observe that the maps $\phi$ and $\psi$ satisfy the inequalities $\phi(w) \leq w \leq \psi(w)$. Furthermore, the fiber of the map $\phi : \Omega_{n,k} \longrightarrow \Omega^{\doubleprime}_{n,k}$ is isomorphic to a Boolean algebra, that is, $\phi^{-1}(v) \cong [v,\psi(v)]$. \end{proof} For $v \in \Omega^{\doubleprime}_{n,k}$ define the statistic $$ \ascodd(v) = |\{ i \: : \: v_{i} < v_{i+1}, i \text{ odd}\}| , $$ that is, $\ascodd(\cdot)$ enumerates the number of ascents in odd positions. \begin{corollary} The $q$-binomial is given by \begin{equation} \label{equation_smaller_q_binomial} \Gaussian{n}{k} = \sum_{v \in \Omega^{\doubleprime}_{n,k}} q^{\inv(v)} \cdot (1+q)^{\ascodd(v)} . \end{equation} \label{corollary_q_1_q} \end{corollary} \begin{proof} It is enough to observe that the sum of the inversion statistic over the elements in the fiber $\phi^{-1}(v) = [v,\psi(v)]$ for $v \in \Omega^{\doubleprime}_{n,k}$ is given by $q^{\inv(v)} \cdot (1+q)^{\ascodd(v)}$. \end{proof} A geometric way to understand this $q$-$(1+q)$-interpretation is to consider lattice paths from the origin $(0,0)$ to $(n-k,k)$ which only use east steps $(1,0)$ and north steps $(0,1)$. Color the squares of this $(n-k) \times k$ board as a chessboard, where the square incident to the origin is colored white. The map $\phi$ in the proof of Theorem~\ref{theorem_decomposition} corresponds to taking a lattice path where every time there is a north step followed by an east step that turns around a white square, we exchange these two steps. The statistic $\ascodd$ enumerates the number of times an east step is followed by a north step when this pair of steps borders a white square. Let $\er(n,k)$ denote the cardinality of the set $\Omega^{\doubleprime}_{n,k}$. Then we have \begin{proposition} The cardinalities $\er(n,k)$ satisfy the recursion \begin{align*} \er(n,k) & = \er(n-2,k-2) + \er(n-2,k-1) + \er(n-2,k) \:\:\:\: \text{ for } 0 \leq k \leq n \text{ and } n \geq 2, \end{align*} with the boundary conditions $\er(0,0) = \er(1,0) = \er(1,1) = 1$ and $\er(n,k) = 0$ whenever $k > n$, $k<0$ or $n < 0$. \end{proposition} \begin{proof} A word in $\Omega^{\doubleprime}_{n,k}$ begins with either $00$, $01$ or $11$, yielding the three cases of the recursion. \end{proof} Directly we obtain the generating polynomial. \begin{theorem} The generating polynomial for $\er(n,k)$ is given by $$ \sum_{k=0}^{n} \er(n,k) \cdot x^{k} = (1+x+x^{2})^{\lfloor n/2 \rfloor} \cdot (1+x)^{n - 2 \cdot \lfloor n/2 \rfloor} . $$ \end{theorem} We end with a statement concerning the symmetry of the $q$-$(1+q)$-binomial. \begin{corollary} The set of defining elements for the $q$-$(1+q)$-binomial satisfy the following symmetric relation: $$ |\Omega_{n,k}''| = |\Omega_{n,n-k}''| . $$ \label{corollary_symmetry} \end{corollary} \begin{proof} This follows from the fact that the generating polynomial for $\er(n,k)$ is a product of palindromic polynomials, and thus is itself is a palindromic polynomial. \end{proof} \section{Analysis of the Fu--Reiner--Stanton--Thiem interpretation} A {\em weak partition} is a finite non-decreasing sequence of non-negative integers. A weak partition~$\lambda = (\lambda_{1}, \ldots, \lambda_{n-k})$ with $n-k$ parts and each part at most $k$ where $\lambda_1 \leq \cdots \leq \lambda_{n-k}$ corresponds to a Ferrers diagram lying inside an $(n-k) \times k$ rectangle with column $i$ having height~$\lambda_i$. These weak partitions are in direct correspondence with the set $\Omega_{n,k}$. Fu, Reiner, Stanton, and Thiem used a pairing algorithm to determine a subset $\Omega_{n,k}' \subseteq \Omega_{n,k}$ of $0$-$1$-sequences to define their $q$-$(1+q)$-analogue of the $q$-binomial; see~\cite[Proposition~6.1]{Fu_Reiner_Stanton_Thiem}. This translates into the following statement. The set $\Omega_{n,k}'$ is in bijection with weak partitions into $n-k$ parts with each part at most $k$ such that \begin{itemize} \item[(a)] if $k$ is even, each odd part has even multiplicity, \item[(b)] if $k$ is odd, each even part (including $0$) has even multiplicity. \end{itemize} \begin{definition} Let $\frst(n,k)$ be the cardinality of the set $\Omega_{n,k}'$. \end{definition} \begin{lemma} The quantity $\frst(n,k)$ counts the number of weak partitions into $n-k$ parts where each part is at most $k$ and each odd part has even multiplicity. \label{lemma_complement} \end{lemma} \begin{proof} When $k$ is even there is nothing to prove. When $k$ is odd, by considering the complement of weak partitions with respect to the rectangle of size $(n-k) \times k$, we obtain a bijective proof. The same complement proof also shows the case when $k$ is even holds. \end{proof} \begin{theorem} The $\frst$-coefficients satisfy the recursion $$ \frst(n,k) = \begin{cases} \frst(n-1,k-1) + \frst(n-1,k), & \text{ if $k$ is even;} \\ \frst(n-2,k-2) + \frst(n-2,k-1) + \frst(n-2,k), & \text{ if $k$ is odd;} \end{cases}$$ where $0 \leq k \leq n$ and $n \geq 2$ with the boundary conditions $\frst(0,0) = \frst(1,0) = \frst(1,1) = 1$ and $\frst(n,k) = 0$ whenever $k > n$, $k<0$ or $n < 0$. \end{theorem} \begin{proof} We use the characterization in Lemma~\ref{lemma_complement}. When $k$ is even there are two cases. If the last part is $k$, remove it to obtain a weak partition counted by $\frst(n-1,k)$. If the last part is less than~$k$, then the weak partition is counted by $\frst(n-1,k-1)$. When $k$ is odd there are three cases. If the last two parts are equal to $k$, then removing these two parts yields a weak partition counted by $\frst(n-2,k)$. Note that we cannot have the last part equal to $k$ and the next to last part less than $k$ since $k$ is odd. If the last part is equal to $k-1$, we can remove it to obtain a weak partition counted by $\frst(n-2,k-1)$. Finally, if the last part is less than or equal to $k-2$, the weak partition is counted by $\frst(n-2,k-2)$. \end{proof} \begin{remark} {\rm For $k$ odd we have the shorter recursion $\frst(n,k) = \frst(n-1,k-1) + \frst(n-2,k)$. However, we use the longer recursion in the proof of Theorem~\ref{theorem_compact}. } \end{remark} \begin{lemma} The inequality $\frst(n,k) \leq \frst(n+1,k+1)$ holds. \label{lemma_frst_inequality} \end{lemma} \begin{proof} The weak partitions which lie inside the rectangle $(n-k) \times k$ and satisfy the conditions of Lemma~\ref{lemma_complement} are included among the weak partitions which lie inside the larger rectangle $(n-k) \times (k+1)$ and satisfy the same conditions. \end{proof} \begin{theorem} For all $0 \leq k \leq n$ the inequality $|\Omega_{n,k}''| = \er(n,k) \leq \frst(n,k) = |\Omega_{n,k}'|$ holds. \label{theorem_compact} \end{theorem} \begin{proof} We proceed by induction on $n$. The induction base is $n \leq 3$. Furthermore, the inequality holds when $k$ is $0$, $1$, $n-1$ and $n$. When $k$ is odd we have that \begin{align*} \er(n,k) & = \er(n-2,k-2) + \er(n-2,k-1) + \er(n-2,k) \\ & \leq \frst(n-2,k-2) + \frst(n-2,k-1) + \frst(n-2,k) \\ & = \frst(n,k) . \end{align*} Similarly, when $k$ is even we have \begin{align*} \er(n,k) & = \er(n-2,k-2) + \er(n-2,k-1) + \er(n-2,k) \\ & \leq \frst(n-2,k-2) + \frst(n-2,k-1) + \frst(n-2,k) \\ & \leq \frst(n-1,k-1) + \frst(n-2,k-1) + \frst(n-2,k) \\ & = \frst(n-1,k-1) + \frst(n-1,k) \\ & = \frst(n,k) , \end{align*} where the second inequality follows from Lemma~\ref{lemma_frst_inequality}. These two cases complete the induction hypothesis. \end{proof} See Table~\ref{table_frst_er} to compare the values of $\frst(n,k)$ and $\er(n,k)$ for $n \leq 10$. \begin{table} $$ \begin{array}{r r r r r r r r r r r c r r r r r r r r r r r} 1 &&&&&&&&&&& \hspace*{5 mm} & 1 \\ 1 & 1 &&&&&&&&&&& 1 & 1 \\ 1 & 1 & 1 &&&&&&&&&& 1 & 1 & 1 \\ 1 & 2 & 2 & 1 &&&&&&&&& 1 & 2 & 2 & 1 \\ 1 & 2 & 4 & 2 & 1 &&&&&&&& 1 & 2 & 3 & 2 & 1 \\ 1 & 3 & 6 & 5 & 3 & 1 &&&&&&& 1 & 3 & 5 & 5 & 3 & 1 \\ 1 & 3 & 9 & 8 & 8 & 3 & 1 &&&&&& 1 & 3 & 6 & 7 & 6 & 3 & 1 \\ 1 & 4 & 12 & 14 & 16 & 9 & 4 & 1 &&&&& 1 & 4 & 9 & 13 & 13 & 9 & 4 & 1 \\ 1 & 4 & 16 & 20 & 30 & 19 & 13 & 4 & 1 &&&& 1 & 4 & 10 & 16 & 19 & 16 & 10 & 4 & 1 \\ 1 & 5 & 20 & 30 & 50 & 39 & 32 & 14 & 5 & 1 &&& 1 & 5 & 14 & 26 & 35 & 35 & 26 & 14 & 5 & 1 \\ 1 & 5 & 25 & 40 & 80 & 69 & 71 & 36 & 19 & 5 & 1 && 1 & 5 & 15 & 30 & 45 & 51 & 45 & 30 & 15 & 5 & 1 \end{array} $$ \caption{The $\frst$- and $\er$-triangles for $n \leq 10$.} \label{table_frst_er} \end{table} \section{Concluding remarks} Is it possible to find a $q$-$(1+q)$-analogue of the Gaussian coefficient which has the smallest possible index set? We believe that our analogue is the smallest, but cannot offer a proof of a minimality. Perhaps a more tractable question is to prove that the Boolean algebra decomposition of $\Omega_{n,k}$ is minimal. We can extend these ideas involving a Boolean algebra decomposition to any distributive lattice. Let~$P$ be a finite poset and let $A$ be an antichain of $P$ such that there is no cover relation in $A$, that is, there is no pair of elements $u, v \in A$ such that $u \prec v$. We obtain a Boolean algebra decomposition of the Birkhoff transform $J(P)$ by defining $$ J^{\doubleprime}(P) = \{I \in J(P) \:\: : \:\: \text{ the ideal $I$ has no maximal elements in the antichain $A$}\} . $$ The two maps $\phi$ and $\psi$ are now defined as \begin{align*} \phi(I) & = I - \{a \in A \: : \: \text{ the element $a$ is maximal in $I$}\} , \\ \psi(I) & = I \cup \{a \in A \: : \: I \cup \{a\} \in J(P)\} . \end{align*} We have the following decomposition theorem. \begin{theorem} For $P$ any finite poset the distributive lattice $J(P)$ has the Boolean algebra decomposition $$ J(P) = \bigcup_{I \in J^{\doubleprime}(P)} [I, \psi(I)] . $$ \end{theorem} Yet again, how can we select the antichain $A$ such that the above decomposition $A$ has the fewest possible terms? Furthermore, would this give the smallest Boolean algebra decomposition? Another way to extend the ideas of Theorem~\ref{theorem_decomposition} is as follows. Define $\Omega^{r}_{n,k}$ to be the set of all words $v \in \Omega_{n,k}$ satisfying the inequalities \begin{align*} & v_{1} \leq v_{2} \leq \cdots \leq v_{r}, \\ & v_{r+1} \leq v_{r+2} \leq \cdots \leq v_{2r}, \\ & \ldots, \\ & v_{r \cdot \lfloor n/r \rfloor - r+1} \leq v_{r \cdot \lfloor n/r \rfloor - r+2} \leq \cdots \leq v_{r \cdot \lfloor n/r \rfloor} . \end{align*} For $1 \leq i \leq \lfloor r/2\rfloor$ define the statistics $b_{i}(v)$ for $v \in \Omega^{r}_{n,k}$ to be $$ b_{i}(v) = |\{j \in [\lfloor n/r \rfloor] \: : \: v_{r j - r+1} + v_{r j - r+2} + \cdots + v_{r j} \in \{i,r-i\}\}| . $$ \begin{theorem} The distributive lattice $\Omega_{n,k}$ has the decomposition $$ \Omega_{n,k} = \bigcup_{v \in \Omega^{r}_{n,k}} \Omega_{r,1}^{b_{1}(v)} \times \Omega_{r,2}^{b_{2}(v)} \timesdots \Omega_{r,\lfloor r/2 \rfloor}^{b_{\lfloor r/2 \rfloor}(v)} . $$ \label{theorem_r} \end{theorem} \begin{corollary} The $q$-binomial is given by $$ \Gaussian{n}{k} = \sum_{v \in \Omega^{r}_{n,k}} q^{\inv(v)} \cdot \Gaussian{r}{1}^{b_{1}(v)} \cdot \Gaussian{r}{2}^{b_{2}(v)} \cdots \Gaussian{r}{\lfloor r/2 \rfloor}^{b_{\lfloor r/2 \rfloor}(v)} . $$ \label{corollary_r} \end{corollary} The least complicated case is when $r=3$, where only one term appears in the above poset product. This term is $\Omega_{3,1}$ which is the three element chain $C_{3}$. The associated Gaussian coefficient is $1+q+q^{2}$. Thus Corollary~\ref{corollary_r} could be called a {\em $q$-$(1+q+q^{2})$-analogue} in the case of $r = 3$. As an example, we have $$ \Gaussian{6}{3} = 1 + q \cdot (1+q+q^{2})^{2} + q^{4} \cdot (1+q+q^{2})^{2} + q^{9} . $$ On a poset level this is a decomposition of $J(C_{3} \times C_{3})$ into two one-element posets of rank~$0$ and rank~$9$, and two copies of $C_{3} \times C_{3}$, where one has its minimal element of rank~$1$ and the other of rank~$4$. \section{Acknowledgments} The authors thank the referee for helpful comments. This work was partially supported by a grant from the Simons Foundation (\#206001 to Margaret~Readdy). \newcommand{\journal}[6]{#1, #2, {\it #3} {\bf #4} (#5), #6.} \newcommand{\book}[4]{#1, {\it #2,} #3, #4.} \newcommand{\collection}[6]{#1, #2, #3, in {\it #4}, #5, #6.} \begin{thebibliography}{99} \bibitem{Cai_Readdy} Y.\ Cai and M.\ Readdy, $q$-Stirling numbers: A new view, preprint, \\ \url{http://arxiv.org/abs/1506.03249}, 2015. \bibitem{Fu_Reiner_Stanton_Thiem} S.\ Fu, V.\ Reiner, D.\ Stanton, and N.\ Thiem, The negative $q$-binomial, {\it Electron J.\ Combin.} {\bf 19} (1) (2012), \href{http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i1p36}{Paper \#P36}. \bibitem{MacMahon_1} P. A. MacMahon, The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects, {\it Amer.\ J.\ Math.} {\bf 35} (1913), 281--322. In {\it Collected Papers. Vol.\ I: Combinatorics}, MIT Press, 1978, pp.\ 508--549. \bibitem{MacMahon_collected} P.\ A.\ MacMahon, Two applications of general theorems in combinatory analysis, {\em Proc.\ London Math.\ Soc.} {\bf 15} (1916), 314--321. In {\it Collected Papers. Vol.\ I: Combinatorics}, MIT Press, 1978, pp.\ 556--563. \bibitem{MacMahon_book} \book{P.\ A.\ MacMahon} {Combinatory Analysis} {Chelsea Publishing Co.} {1960} \bibitem{EC1} R. P. Stanley, {\it Enumerative Combinatorics}, Vol.~1, 2nd edition. Cambridge University Press, 2012. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 06A07; Secondary 05A05, 05A10, 05A30. \noindent \emph{Keywords:} $q$-analogue, Birkhoff transform, distributive lattice, poset decomposition. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received May 19 2016; revised versions received August 31 2016; September 8 2016. Published in {\it Journal of Integer Sequences}, September 11 2016. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}