\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amsmath,amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage{amsthm} \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{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf Partition Statistics and $q$-Bell Numbers ($q=-1$)} \vskip 1cm \large Carl G. Wagner\\ Department of Mathematics\\ University of Tennessee\\ Knoxville, TN 37996-1300, USA\\ \href{wagner@math.utk.edu}{\tt wagner@math.utk.edu} \\ \end{center} \vskip .2 in \begin{abstract} We study three partition statistics and the $q$-Stirling and $q$-Bell numbers that serve as their generating functions, evaluating these numbers when $q=-1$. Among the numbers that arise in this way are (1) Fibonacci numbers and (2) numbers occurring in the study of fermionic oscillators. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{lemma}{Lemma}[section] %\usepackage{amsmath} %\usepackage{amssymb} %\usepackage{amscd} %\usepackage{amsthm} %\numberwithin{equation}{section} %\theoremstyle{plain} %\newtheorem{theo}{Theorem} %\numberwithin{theo}{section} \section{Introduction}\label{sec1} The notational conventions of this paper are as follows: $\mathbb{N} :=\{0,1,2,\dots\}$, $\mathbb{P}:=\{1,2,\dots\}$, $[0]:=\varnothing$, and $[n]:=\{1,\dots,n\}$ for $n\in\mathbb{P}$. Empty sums take the value $0$ and empty products the value $1$, with $0^0:=1$. The letter $q$ denotes an indeterminate, with $0_q:=0$, $n_q:=1+q+\dots+q^{n-1}$ for $n\in\mathbb{P}$, $0\mathop{!}\limits_q:=1$, and $n\mathop{!}\limits_q:=1_q2_q\cdots n_q$ for $n\in\mathbb{P}$. The binomial coefficient $\binom nk$ is equal to zero if $k$ is a negative integer or if $0\leqslant ni_s$ \cite[Corollary 1.3.10]{bSt}. Of course, $G(I,\Delta;1)=|\Delta|$. On the other hand, \begin{equation}\label{e1.4} G(I,\Delta;-1)=\left|\{\delta\in\Delta:\ I(\delta)\text{ is even}\}\right| -\left|\{\delta\in\Delta:\ I(\delta)\text{ is odd}\}\right|. \end{equation} Hence if $G(I,\Delta;-1)=0$, the set $\Delta$ is ``balanced'' with respect to the parity of $I$. In particular, setting $q=-1$ in \eqref{e1.2} yields the familiar result that a finite nonempty set has as many subsets of odd cardinality as it has subsets of even cardinality. Setting $q=-1$ in \eqref{e1.3} reveals that if $n\geqslant2$, then among the permutations of $[n]$ there are as many with an odd number of inversions as there are with an even number of inversions. In this note we consider three $q$-generalizations of Stirling numbers of the second kind, denoted $S_q^*(n,k)$, $S_q(n,k)$, and $\tilde S_q(n,k)$. These polynomials are generating functions for three closely related statistics on the set of partitions of $[n]$ with $k$ blocks. Most of the properties of these $q$-Stirling numbers, to be established below in \S~\ref{sec3}, have appeared in the literature in various contexts, Carlitz \cite{bCa} having apparently been the first to construe these numbers as generating functions for partition statistics. See also \cite{bESW}, \cite{bMi}, \cite{bWa}, and \cite{bWW}. Our aim here is to offer a compact, unified treatment of these numbers. Our analysis is facilitated by a powerful formal algebraic result of Comtet \cite{bCo}. We derive in \S~\ref{sec4} new results on the evaluation of $S_q^*(n,k)$, $S_q(n,k)$, and $\tilde S_q(n,k)$ and their associated $q$-Bell numbers (gotten by summing $q$-Stirling numbers over $k$ for fixed $n$) when $q=-1$. Apart from the interpretation of these results in terms of \eqref{e1.4}, the evaluation of $S_{-1}(n,k)$ and its associated Bell numbers may be of additional interest, since these numbers arise in the study of fermionic oscillators \cite{bSc}. In \S~\ref{sec5} we discuss an alternative approach to establishing our results by means of bijective proofs. In \S~\ref{sec6} the numbers $S_{-1}^*(n,k)$, $S_{-1}(n,k)$, and $\tilde S_{-1}(n,k)$ are displayed as triangular arrays for $1\leqslant k\leqslant n\leqslant 8$. Here, for immediate reference, we record these arrays in linearized form: \[ \begin{aligned} S_{-1}^*(n,k) &= \mbox{\small$\begin{array}[t]{rrrrrrrrrrrrr} \!-1,&\!1,&\!-1,&\!-1,&\!1,&\!1,&\!1,&\!-1,&\!-2,&\!1,&\!-1,&\!1,\\ \!3,&\!-2,&\!-1,&\!1,&\!-1,&\!-4,&\!3,&\!3,&\!-1,&\!-1,&\!1,&\!5,\\ \!-4,&\!-6,&\!3,&\!1,&1,&\!-1,&\!-6,&\!5,&\!10,&\!-6,&\!-4,&\!1,&\!\dots\end{array}$}\\ S_{-1}(n,k) &= \mbox{\small$\begin{array}[t]{rrrrrrrrrrrrr} \!1,&\!1,&\!-1,&\!1,&\!-1,&\!-1,&\!1,&\!-1,&\!-2,&1,&\!1,&\!-1,\\ \!-3,&\!2,&\!1,&\!1,&\!-1,&\!-4,&\!3,&\!3,&-1,&\!1,&\!-1,&\!-5,\\ \!4,&\!6,&\!-3,&\!-1,&\!1,&\!-1,&\!-6,&\!5,&10,&\!-6,&\!-4,&\!1,&\!\dots \end{array}$}\\ \tilde S_{-1}(n,k) &= \mbox{\small$\begin{array}[t]{rrrrrrrrrrrrr} 1,&1,&1,&1,&1,&1,&1,&1,&2,&1,&1,&1,\\ 3,&2,&1,&1,&1,&4,&3,&3,&1,&1,&1,&5,\\ 4,&6,&3,&1,&1,&1,&6,&5,&10,&6,&4,&1,&\dots \end{array}$} \end{aligned} \] \section{Preliminaries}\label{sec2} This section reviews some material to be used later in the paper. 2.1. {\em Comtet Numbers}. The following theorem, due to Comtet \cite{bCo} greatly facilitates the analysis of many combinatorial arrays: \begin{theorem}\label{th2.1} Let $D$ be an integral domain. If $(u_n)_{n\geqslant0}$ is a sequence in $D$ and $x$ is an indeterminate over $D$, then the following are equivalent characterizations of an array $(U(n,k))_{n,k\geqslant0}$: \begin{gather} \label{e2.3} U(n,k)=U(n-1,k-1)+u_kU(n-1,k),\qquad \forall\ n,k\in\mathbb{P},\\ \intertext{with $U(n,0)=u_0^n$ and $U(0,k)=\delta_{0,k}$ $\forall$ $n,k\in\mathbb{N}$,} \label{e2.1} U(n,k)=\sum_{\substack{d_0+d_1+\dots+d_k=n-k\\ d_i\in\mathbb{N}}} u_0^{d_0}u_1^{d_1}\cdots u_k^{d_k},\qquad \forall\ n,k\in\mathbb{N},\\ \label{e2.2} \sum_{n\geqslant0}U(n,k)x^n=\frac{x^k}{(1-u_0x)(1-u_1x)\cdots(1-u_kx)},\qquad \forall\ k\in\mathbb{N},\\ \intertext{and} \label{e2.4} x^n=\sum_{k=0}^nU(n,k)p_k(x),\qquad\forall\ n\in\mathbb{N}, \end{gather} where $p_0(x):=1$ and $p_k(x):=(x-u_0)\cdots(x-u_{k-1})$ for $k\in\mathbb{P}$. \end{theorem} \begin{proof}[\bf Proof] Straightforward algebraic exercise. \end{proof} In what follows, we call the numbers $U(n,k)$ the {\em Comtet numbers associated with the sequence} $(u_n)_{n\geqslant0}$. 2.2 {\em Partitions of a Set}. A {\em partition} of a set $S$ is a set of nonempty, pairwise disjoint subsets (called {\em blocks}) of $S$, with union $S$. For all $n,k\in\mathbb{N}$, let $S(n,k)$ denote the number of partitions of $[n]$ with $k$ blocks. Then $S(0,0)=1$, $S(n,0)=S(0,k)=0$, $\forall$ $n,k\in\mathbb{P}$, and \begin{equation}\label{e2.5} S(n,k)=S(n-1,k-1)+kS(n-1,k),\qquad\forall\ n,k\in\mathbb{P}, \end{equation} $S(n-1,k-1)$ enumerating those partitions in which $n$ is the sole element of one of the blocks, and $kS(n-1,k)$ those in which the block containing $n$ contains at least one other element of $[n]$. From \eqref{e2.5} it follows that the numbers $S(n,k)$, called {\em Stirling numbers of the second kind}, are the Comtet numbers associated with the sequence $(0,1,2,\dots)$. Hence by Theorem \ref{th2.1} \begin{gather*} %\label{e2.6} S(n,k)=\sum_{\substack{d_1+\dots+d_k=n-k\\ d_i\in\mathbb{N}}}1^{d_1}2^{d_2} \cdots k^{d_k},\qquad\forall\ n,k\in\mathbb{N},\\ %\label{e2.7} \sum_{n\geqslant0}S(n,k)x^n=\frac{x^k}{(1-x)(1-2x)\cdots(1-kx)},\qquad \forall\ k\in\mathbb{N},\\ \intertext{and} %\label{e2.8} x^n=\sum_{k=0}^nS(n,k)x^{\underbar{$\scriptstyle k$}},\qquad\forall\ n\in\mathbb{N}, \end{gather*} where $x^{\underbar{$\scriptstyle0$}}:=1$ and $x^{\underbar{$\scriptstyle k$}}:=x(x-1)\cdots(x-k+1)$ for $k\in\mathbb{P}$. The total number of partitions of $[n]$ is given by the {\em Bell number} $B_n$, where \begin{equation*}%\label{e2.9} B_n=\sum_{k=0}^nS(n,k). \end{equation*} Clearly, $B_0=1$, and \begin{equation*}%\label{e2.10} B_{n+1}=\sum_{k=0}^n\binom nkB_k, \end{equation*} since $\binom nkB_k$ enumerates those partitions of $[n+1]$ for which the size of the block containing the element $n+1$ is $n-k+1$. 2.3 {\em Restricted Sums of Binomial Coefficients}. As we have already noted in \S~\ref{sec1}, setting $q=1$ and $q=-1$ in \eqref{e1.2} yields the well known result \begin{equation*}%\label{e2.11} \sum_{k\text{ even}}\binom nk=\sum_{k\text{ odd}}\binom nk=2^{n-1},\qquad \forall\ n\in\mathbb{P}. \end{equation*} Here we recall a method for evaluating sums such as \begin{equation*}%\label{e2.12} \sum_{k\equiv0\!\!\!\pmod3}\binom nk. \end{equation*} Let $\omega$ be either of the two complex cube roots of $1$, e.g., $\omega=(-1+i\sqrt3)/2$. Then \begin{multline}\label{e2.13} (1+x)^n+(1+\omega x)^n+(1+\omega^2x)^n=\sum_{k=0}^n\binom nkx^k\left(1+\omega^k+\omega^{2k}\right)\\ =3\sum_{k\equiv0\!\!\!\pmod3}\binom nkx^k, \end{multline} since $k\equiv0\pmod3$ implies that $1+\omega^k+\omega^{2k}=3$ and $k\equiv1$ or $2$ $\pmod 3$ implies that $1+\omega^k+\omega^{2k}=1+\omega+\omega^2=0$. Setting $x=1$ in \eqref{e2.13} yields \begin{equation}\label{e2.14} \sum_{k\equiv0\!\!\!\pmod3}\binom nk=\frac13\left(2^n+(1+\omega)^n+(1+\omega^2)^n\right). \end{equation} \section{Partition Statistics and $q$-Stirling Numbers}\label{sec3} Let $\Pi(n,k)$ denote the set of all partitions of $[n]$ with $k$ blocks. Given a partition $\pi\in\Pi(n,k)$, let $(E_1,\dots,E_k)$ be the unique {\em ordered} partition of $[n]$ comprising the same blocks as $\pi$, arranged in increasing order of their smallest elements, and define statistics $w^*$, $w$, and $\tilde w$ by \begin{align*} %\label{e3.1} w^*(\pi) &:= \sum_{i=1}^ki|E_i|,\\ %\label{e3.2} w(\pi) &:= \sum_{i=1}^k(i-1)|E_i|=w^*(\pi)-n,\\ \intertext{and} %\label{e3.3} \tilde w(\pi) &:=\sum_{i=1}^k(i-1)(|E_i|-1)=w^*(\pi)-n-\binom k2. \end{align*} If elements of $[n]$ are regarded as labels on $n$ unit masses, then $w^*(\pi)$ is the moment about $x=0$ of the mass configuration in which the masses with labels in $E_i$ are placed at $x=i$. The statistics $w(\pi)$ and $\tilde w(\pi)$ admit of similar interpretations. We wish to study the generating functions \begin{align} \label{e3.4} S_q^*(n,k) &:= \sum_{\pi\in\Pi(n,k)}q^{w^*(\pi)},\\ \label{e3.5} S_q(n,k) &:= \sum_{\pi\in\Pi(n,k)}q^{w(\pi)}=q^{-n}S_q^*(n,k),\\ \intertext{and} \label{e3.6} \tilde S_q(n,k) &:= \sum_{\pi\in\Pi(n,k)}q^{\tilde w(\pi)} =q^{-n-\binom k2}S_q^*(n,k). \end{align} Each of these polynomials furnishes a $q$-generalization of $S(n,k)$, reducing to the latter when $q=1$. As closely related as these $q$-Stirling numbers appear to be, it might be thought that one could carry out an analysis of any one of them, chosen arbitrarily, with properties of the others derived as easy corollaries. Interestingly, it turns out that each is best suited for elucidating a particular subset of their more or less common properties. We consider first the matter of recursive formulas. \begin{theorem}\label{th3.1} The $q$-Stirling numbers $S_q^*(n,k)$ are generated by the recurrence relation \begin{equation}\label{e3.7} S_q^*(n,k)=q^kS_q^*(n-1,k-1)+qk_qS_q^*(n-1,k),\qquad\forall\ n,k\in\mathbb{P}, \end{equation} with $S_q^*(0,0)=1$ and $S_q^*(n,0)=S_q^*(0,k)=0$, $\forall$ $n,k\in \mathbb{P}$. \end{theorem} \begin{proof}[\bf Proof] The boundary conditions are obvious. To establish the recurrence \eqref{e3.7}, let \[ c(n,k,t):=\left|\{\pi\in\Pi(n,k):\ w^*(\pi)=t\}\right|. \] Then, \begin{equation}\label{e3.8} c(n,k,t)=c(n-1,k-1,t-k)+\sum_{i=1}^kc(n-1,k,t-i),\qquad\forall\ n,k\in \mathbb{P}. \end{equation} For if $w^*(\pi)=t$, with $(E_1,\dots,E_k)$ being the ordered partition associated with $\pi$, then the number $n\in[n]$ is either (i) in $E_k$ alone (there are clearly $c(n-1,k-1,t-k)$ such $\pi$'s) or (ii) in some $E_i$, where $1\leqslant i\leqslant k$, with at least one element of $[n-1]$ (there are clearly $c(n-1,k,t-i)$ such $\pi$'s). From \eqref{e3.8} it follows that \[ \begin{split} S_q^*(n,k) &= \sum_tc(n,k,t)q^t\\ &= \sum_rc(n-1,k-1,r)q^{r+k}+\sum_{i=1}^kq^i\sum_rc(n-1,k,r)q^r\\ &= q^kS_q^*(n-1,k-1)+qk_qS_q^*(n-1,k). \end{split} \] \end{proof} Recurrence relations for $S_q(n,k)$ and $\tilde S_q(n,k)$ follow immediately from \eqref{e3.7}, along with \eqref{e3.5} and \eqref{e3.6}, respectively. We have \begin{align} \label{e3.9} S_q(n,k)&=q^{k-1}S_q(n-1,k-1)+k_qS_q(n-1,k),\qquad\forall\ n,k\in\mathbb{P},\\ \intertext{and} \label{e3.10} \tilde S_q(n,k) &= \tilde S_q(n-1,k-1)+k_q\tilde S_q(n-1,k),\qquad\forall\ n,k\in\mathbb{P}. \end{align} By \eqref{e3.10}, the numbers $\tilde S_q(n,k)$ are the Comtet numbers associated with the sequence $(n_q)_{n\geqslant0}$. By Theorem \ref{th2.1} it follows immediately that \begin{gather} \label{e3.11} \tilde S_q(n,k)=\sum_{\substack{d_1+\dots+d_k=n-k\\ d_i\in\mathbb{N}}} (1_q)^{d_1}(2_q)^{d_2}\cdots(k_q)^{d_k},\qquad\forall\ n,k\in\mathbb{N},\\ \label{e3.12} \sum_{n\geqslant0}\tilde S_q(n,k)x^n=\frac{x^k}{(1-1_qx)(1-2_qx)\cdots (1-k_qx)},\qquad\forall\ k\in\mathbb{N},\\ \intertext{and} \label{e3.13} x^n=\sum_{k=0}^n\tilde S_q(n,k)\phi_k(x),\qquad\forall\ n\in\mathbb{N}, \end{gather} where $\phi_0(x):=1$ and $\phi_k(x):=x(x-1_q)\cdots(x-(k-1)_q)$, $\forall$ $k\in\mathbb{P}$. Variants of \eqref{e3.11}--\eqref{e3.13} that hold for $S_q(n,k)$ and $S_q^*(n,k)$ follow immediately from the relations $S_q(n,k)=q^{\binom k2} \tilde S_q(n,k)$ and $S_q^*(n,k)=q^nS_q(n,k)$. To cite a few examples, we have \begin{multline*} %\label{e3.14} \sum_{n\geqslant0}S^*(n,k)x^n= \frac{q^{\binom{k+1}2}x^k}{(1-qx)(1-qx-q^2x)\cdots(1-qx-\dots-q^kx)},\\ %\qquad \forall\ k\in\mathbb{N}, \end{multline*} and \begin{equation} \label{e3.15} x^n=\sum_{k=0}^nS_q(n,k)\psi_k(x)=\sum_{k=0}^nS_q^*(n,k)\psi_k\left( \frac xq\right), \end{equation} where $\psi_k(x):=q^{-\binom k2}\phi_k(x)$. Using the method of linear functionals \cite[pp. 89--90]{bRo} one can derive from \eqref{e3.15} the recurrence \cite[Theorem 5.4]{bWa} \begin{equation}\label{e3.16} S_q(n+1,k)=\sum_{j=0}^n\binom njq^jS_q(j,k-1),\qquad \forall\ n\in\mathbb{N},\ k\in\mathbb{P}, \end{equation} from which the variant recurrences \begin{align} %\label{e3.17} \notag S_q^*(n+1,k) &= q^{n+1}\sum_{j=0}^n\binom nj S_q^*(j,k-1),\qquad \forall\ n\in\mathbb{N},\ k\in\mathbb{P},\\ \intertext{and} \label{e3.18} \tilde S_q(n+1,k) &= \sum_{j=0}^n\binom nj q^{j-k+1} \tilde S_q(j,k-1),\qquad\forall\ n\in\mathbb{N},\ k\in\mathbb{P} \end{align} follow immediately. Summing the $q$-Stirling numbers $S_q^*(n,k)$, $S_q(n,k)$ and $\tilde S_q(n,k)$ over $k$ yields the respective $q$-Bell numbers $B_q^*(n)$, $B_q(n)$, and $\tilde B_q(n)$. From \eqref{e3.16} it follows that \begin{equation}\label{e3.19} B_q(n+1)=\sum_{j=0}^n\binom njq^jB_q(j),\qquad\forall\ n\in\mathbb{N}. \end{equation} Since $B_q^*(n)=q^nB_q(n)$, the recurrence \eqref{e3.19} yields \begin{equation}\label{e3.20} B_q^*(n+1)=q^{n+1}\sum_{j=0}^n\binom njB_q^*(j),\qquad\forall\ n\in \mathbb{N}. \end{equation} Due to the factor $q^{-k}$ in \eqref{e3.18}, we do not get any recurrence for $\tilde B_q(n)$ analogous to \eqref{e3.19} and \eqref{e3.20}, this being the single exception to the general parallelism between properties of the three $q$-Stirling numbers under consideration. The uniqueness of $\tilde B_q(n)$ is further manifested when $q=-1$, as we shall see in the next section. \section{The Case $q=-1$}\label{sec4} In this section we derive simple expressions for the foregoing $q$-Stirling and $q$-Bell numbers when $q=-1$. \begin{theorem}\label{th4.1} The number $\tilde S_{-1}(n,k)$ is given by the formula \begin{equation} \label{e4.1} \tilde S_{-1}(n,k)=\binom{n-\left\lfloor \frac k2\right\rfloor-1}{\left\lfloor\frac{k-1}2\right\rfloor},\qquad 1\leqslant k\leqslant n. \end{equation} \end{theorem} \begin{proof}[\bf Proof] Note that \begin{equation*}%\label{e4.4} \left.i_q\right|_{q=-1}=\omega_i:= \begin{cases} 1, &\text{if $i$ is odd};\\ 0, &\text{if $i$ is even}. \end{cases} \end{equation*} Hence by \eqref{e3.11}, if $1\leqslant m\leqslant\lfloor n/2\rfloor$, \begin{equation}\label{e4.5} \tilde S_{-1}(n,2m)=\sum_{\substack{d_1+d_3+\dots+d_{2m-1}=n-2m\\ d_i\in\mathbb{N}}}1=\binom{n-m-1}{m-1}, \end{equation} since the number of sequences $(t_1,\dots, t_m)$ of nonnegative integers summing to $s$ is $\binom {s+m-1}{m-1}$ \cite[p. 15]{bSt}. Similarly, if $0\leqslant m\leqslant\lfloor (n-1)/2\rfloor$, \begin{equation} \label{e4.2} \tilde S_{-1}(n,2m+1)=\binom{n-m-1}m. \end{equation} Formula \eqref{e4.1} incorporates \eqref{e4.5} and \eqref{e4.2}. \end{proof} In tabulating the numbers $\tilde S_{-1}(n,k)$ it is of course more efficient to use the recurrence \begin{equation*}%\label{e4.6} \tilde S_{-1}(n,k)=\tilde S_{-1}(n-1,k-1)+\omega_k\tilde S_{-1}(n-1,k), \end{equation*} representing the case $q=-1$ of \eqref{e3.10}. Let $F_0=F_1=1$, with $F_n=F_{n-1}+F_{n-2}$ if $n\geqslant2$. As is well known, \begin{equation}\label{eq4.6} F_n=\sum_{m=0}^{\lfloor n/2\rfloor}\binom{n-m}m,\qquad \forall n\in\mathbb{N}. \end{equation} \begin{theorem}\label{theo4.2} For all $n\in\mathbb{N}$, \begin{equation}\label{eq4.7} \tilde B_{-1}(n):=\sum_{k=0}^n\tilde S_{-1}(n,k)=F_n. \end{equation} \end{theorem} \begin{proof}[\bf Proof] It is easy to check that \eqref{eq4.7} holds for $n=0,1$. If $n\geqslant2$, then by \eqref{e4.5}, \eqref{e4.2}, and \eqref{eq4.6}, \[ \begin{split} \tilde B_{-1}(n) &= \sum_{m=0}^{\lfloor (n-1)/2\rfloor}\binom{n-m-1}m +\sum_{m=1}^{\lfloor n/2\rfloor}\binom{n-m-1}{m-1}\\ &= \sum_{m=0}^{\lfloor (n-1)/2\rfloor}\binom{(n-1)-m}m+ \sum_{m=0}^{\lfloor (n-2)/2\rfloor}\binom{(n-2)-m}m\\ &= F_{n-1}+F_{n-2}=F_n. \end{split} \] \end{proof} From \eqref{e4.1} and the fact that $S_q^*(n,k)=q^{\binom k2+n}\tilde S_q(n,k)$, we have \begin{equation*}%\label{eq4.8} S_{-1}^*(n,k)=(-1)^{\binom k2+n}\binom{n-\left\lfloor\frac k2\right\rfloor-1}% {\left\lfloor\frac{k-1}2\right\rfloor},\qquad 1\leqslant k\leqslant n. \end{equation*} On the other hand, the Bell numbers $B_{-1}^*(n)$ are quite different from the numbers $\tilde B_{-1}(n)$. \begin{theorem}\label{th4.2} For all $n\in\mathbb{N}$, \begin{equation}\label{e4.9} B_{-1}^*(n):=\sum_{k=0}^nS_{-1}^*(n,k)=\begin{cases} 1, &\text{if\ \ }n\equiv0\pmod3;\\ -1, &\text{if\ \ }n\equiv1\pmod3;\\ 0, &\text{if\ \ }n\equiv2\pmod3. \end{cases} \end{equation} \end{theorem} \begin{proof}[\bf Proof] Noting that $B_{-1}^*(0)=1$, we prove \eqref{e4.9} by induction on $n$. In what follows \begin{equation*}%\label{e4.10} b_r(n):=\sum_{j\equiv r\!\!\!\pmod3}\binom nj. \end{equation*} From \eqref{e3.20} with $q=-1$, we have \begin{multline*}%\label{e4.11} B_{-1}^*(n+1)=(-1)^{n+1}\sum_{j=0}^n\binom njB_{-1}^*(j)=(-1)^{n+1} b_0(n)+(-1)^nb_1(n)\\ =(-1)^{n+1}b_0(n)+(-1)^nb_0(n-1)+(-1)^nb_1(n-1). \end{multline*} Similarly, $B_{-1}^*(n)=(-1)^nb_0(n-1)+(-1)^{n-1}b_1(n-1)$, and so \begin{multline}\label{e4.12} B_{-1}^*(n+1)=(-1)^{n+1}b_0(n)+2(-1)^nb_0(n-1)-B_{-1}^*(n)\\ =\frac13\left[\omega^{2n-1}-\omega^{2n-2}+\omega^{n+1}-\omega^{n-1}\right] -B_{-1}^*(n), \end{multline} by \eqref{e2.14}, where $\omega$ is either of the two complex cube roots of $1$. Taking $n+1=3m$, $3m+1$, and $3m+2$, respectively, in \eqref{e4.12} yields \begin{align*} %\label{e4.13} B_{-1}^*(3m) &= 1-B_{-1}^*(3m-1)=1,\\ %\label{e4.14} B_{-1}^*(3m+1) &= 0-B_{-1}^*(3m)=-1,\qquad\text{and}\\ %\label{e4.15} B_{-1}^*(3m+2) &= -1-B_{-1}^*(3m+1)=0. \end{align*} \end{proof} It is easy to check that one can write \eqref{e4.9} more compactly as \begin{equation*}%\label{e4.16} B_{-1}^*(n)= \frac1{1-\omega}\omega^n-\frac\omega{1-\omega}\omega^{2n}, \end{equation*} from which we get the nice exponential generating function \begin{equation}\label{e4.17} \sum_{n=0}^\infty B_{-1}^*(n)\frac{x^n}{n!}=\frac1{1-\omega}e^{\omega x}- \frac \omega{1-\omega}e^{\omega^2x}. \end{equation} From \eqref{e4.1} and the fact that $S_q(n,k)=q^{\binom k2}\tilde S_q(n,k)$, we have \begin{equation*}%\label{eq4.18} S_{-1}(n,k)=(-1)^{\binom k2}\binom{n-\left\lfloor\frac k2\right\rfloor-1}% {\left\lfloor\frac{k-1}2\right\rfloor},\qquad 1\leqslant k\leqslant n. \end{equation*} By \eqref{e3.5}, \[ B_{-1}(n):=\sum_{k=0}^nS_{-1}(n,k)=(-1)^nB_{-1}^*(n), \] and so by \eqref{e4.9} \begin{equation}\label{e4.18} B_{-1}(n)= \begin{cases} (-1)^n, &\text{if\ \ }n\equiv0\pmod3;\\ (-1)^{n+1}, &\text{if\ \ }n\equiv1\pmod3;\\ 0, &\text{if\ \ }n\equiv2\pmod3, \end{cases} \end{equation} and by \eqref{e4.17} \begin{equation}\label{e4.19} \sum_{n=0}^\infty B_{-1}(n)\frac{x^n}{n!}=\frac1{1-\omega}e^{-\omega x}- \frac \omega{1-\omega}e^{-\omega^2x}. \end{equation} In a paper which showed how the numbers $S_{-1}(n,k)$ and $B_{-1}(n)$ arise in the study of fermionic oscillators, Schork \cite{bSc} posed the problem of finding a closed form and a generating function for the numbers $B_{-1}(n)$. Formulas \eqref{e4.18} and \eqref{e4.19} furnish solutions to Schork's problem. To conclude this section we remark that our list of partition statistics might have been rounded out to include the statistic \begin{equation*}%\label{e4.21} \hat w(\pi):=\sum_{i=1}^k i(|E_i|-1)=\tilde w(\pi)+n-k, \end{equation*} with generating function \begin{equation}\label{e4.22} \hat S_q(n,k):=\sum_{\pi\in\Pi(n,k)}q^{\hat w(\pi)}=q^{(n-k)}\tilde S_q(n,k). \end{equation} Formula \eqref{e4.22} and Theorem \ref{th4.1} yield an easy evaluation of $\hat S_{-1}(n,k)$. As for \begin{equation*}%\label{e4.23} \hat B_{-1}(n):=\sum_{k=0}^n\hat S_{-1}(n,k), \end{equation*} we have $\hat B_{-1}(0)=\hat B_{-1}(1)=1$, $\hat B_{-1}(2)=0$, and \begin{equation}\label{e4.24} \hat B_{-1}(n)=(-1)^{n-1}F_{n-3},\qquad \forall n\geqslant 3, \end{equation} the proof of which we leave to interested readers. \section{Bijective Proofs}\label{sec5} We conclude by returning to the opening theme of this paper. If $G(I,\Delta;-1)\allowbreak=0$, then, as already noted, $|\Delta_0|=|\Delta_1|$, where $\Delta_i=\{\delta\in\Delta:\ I(\delta)\equiv i\pmod 2\}$. One gains a deeper understanding (and bijective proof) of such results by identifying an $I$-parity changing involution of $\Delta$. For the statistic $|S|$ in \eqref{e1.2}, the map \[ S\mapsto\begin{cases} S\cup\{1\}, &\text{if }1\notin S;\\ S-\{1\}, &\text{if }1\in S, \end{cases} \] furnishes such an involution. For the statistic $i(\sigma)$ in \eqref{e1.3}, switching positions of the elements $1$ and $2$ (or of $k$ and $k+1$, for any fixed $k$) in a permutation furnishes such an involution. A similar task arises when \eqref{e1.4} is nonzero. Suppose, for example, that $G(I,\Delta;-1)=c>0$. Here one wishes to identify a subset $\Delta^+$ of $\Delta_0$, with $|\Delta^+|=c$, and an $I$-parity changing involution of $\Delta-\Delta^+$. My student, Mark Shattuck, has recently succeeded in finding such bijective proofs of formulas \eqref{e4.1}, \eqref{eq4.7}, \eqref{e4.9}, \eqref{e4.18}, and \eqref{e4.24}. Details will appear in a forthcoming paper. \section{Tables}\label{sec6} \noindent Table 1: The numbers $S_{-1}^*(n,k)$ for $1\leqslant k\leqslant n \leqslant8$. \begin{center} \begin{tabular}{|r||r|r|r|r|r|r|r|r|} \hline & $k=1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ \\ \hline\hline $n=1$ & $-1$ & & & & & & & \\ \hline $2$ & $1$ & $-1$ & & & & & & \\ \hline $3$ & $-1$ & $1$ & $1$ & & & & & \\ \hline $4$ & $1$ & $-1$ & $-2$ & $1$ & & & & \\ \hline $5$ & $-1$ & $1$ & $3$ & $-2$ & $-1$ & & & \\ \hline $6$ & $1$ & $-1$ & $-4$ & $3$ & $3$ & $-1$ & & \\ \hline $7$ & $-1$ & $1$ & $5$ & $-4$ & $-6$ & $3$ & $1$ & \hphantom{$-1$} \\ \hline $8$ & $1$ & $-1$ & $-6$ & $5$ & $10$ & $-6$ & $-4$ & $1$ \\ \hline \end{tabular} \end{center} \noindent Table 2: The numbers $S_{-1}(n,k)$ for $1\leqslant k\leqslant n \leqslant8$. \begin{center} \begin{tabular}{|r||r|r|r|r|r|r|r|r|} \hline & $k=1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ \\ \hline\hline $n=1$ & $1$ & & & & & & & \\ \hline $2$ & $1$ & $-1$ & & & & & & \\ \hline $3$ & $1$ & $-1$ & $-1$ &\hphantom{$-1$}&\hphantom{$-1$}& & & \\ \hline $4$ & $1$ & $-1$ & $-2$ & $1$ & & & & \\ \hline $5$ & $1$ & $-1$ & $-3$ & $2$ & $1$ & & & \\ \hline $6$ & $1$ & $-1$ & $-4$ & $3$ & $3$ & $-1$ & & \\ \hline $7$ & $1$ & $-1$ & $-5$ & $4$ & $6$ & $-3$ & $-1$ & \hphantom{$-1$} \\ \hline $8$ & $1$ & $-1$ & $-6$ & $5$ & $10$ & $-6$ & $-4$ & $1$ \\ \hline \end{tabular} \end{center} \noindent Table 3: The numbers $\tilde S_{-1}(n,k)$ for $1\leqslant k \leqslant n\leqslant 8$. \begin{center} \begin{tabular}{|r||r|r|r|r|r|r|r|r|} \hline & $k=1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ \\ \hline\hline $n=1$ & $1$ &\hphantom{$-1$}&\hphantom{$-1$}&\hphantom{$-1$}&\hphantom{$-1$}&\hphantom{$-1$}&\hphantom{$-1$}& \\ \hline $2$ & $1$ & $1$ & & & & & & \\ \hline $3$ & $1$ & $1$ & $1$ & & & & & \\ \hline $4$ & $1$ & $1$ & $2$ & $1$ & & & & \\ \hline $5$ & $1$ & $1$ & $3$ & $2$ & $1$ & & & \\ \hline $6$ & $1$ & $1$ & $4$ & $3$ & $3$ & $1$ & & \\ \hline $7$ & $1$ & $1$ & $5$ & $4$ & $6$ & $3$ & $1$ & \hphantom{$-1$} \\ \hline $8$ & $1$ & $1$ & $6$ & $5$ & $10$ & $6$ & $4$ & $1$ \\ \hline \end{tabular} \end{center} \begin{thebibliography}{99} \bibitem{bCa} L. Carlitz, {\rm Generalized Stirling numbers}, {\it Combinatorial Analysis Notes}, Duke University (1968), 8-{}-15. \bibitem{bCo} L. Comtet, {\rm Nombres de Stirling g\'en\'eraux et fonctions symmetriques}, {\it C. R. Acad. Sci. Paris S\'erie A} {\bf 275} (1972), 747-{}-750. \bibitem{bESW} P. Edelman, R. Simion, and D. White, {\rm Partition statistics on permutations}, {\it Discrete Math.} {\bf 99} (1992), 62-{}-68. \bibitem{bMi} S. Milne, {\rm A $q$-analogue of restricted growth functions, Dobinski's equality, and Charlier polynomials}, {\it Trans. Amer. Math. Soc.} {\bf 245} (1978), 89-{}-118. \bibitem{bRo} G.-C. Rota, {\rm The number of partitions of a set}, {\it Amer. Math. Monthly} {\bf 71} (1964), 498-{}-504. \bibitem{bSc} M. Schork, {\rm On the combinatorics of normal ordering bosonic operators and deformations of it}, {\it J. Phys. A: Math. Gen.} {\bf 36} (2003), 4651-{}-4665. \bibitem{bSt} R. Stanley, {\it Enumerative Combinatorics, Vol.} I, Wadsworth and Brooks/Cole, 1986. \bibitem{bWa} C. Wagner, {\rm Generalized Stirling and Lah numbers}, {\it Discrete Math.} {\bf 160} (1996), 199-{}-218. \bibitem{bWW} M. Wachs and D. White, {\rm $p,q$-Stirling numbers and partition statistics}, {\it J. Combin. Theory A} {\bf 56} (1991), 27-{}-46. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11B73; Secondary 11B39. \noindent \emph{Keywords:} partition statistics, $q$-Stirling numbers, $q$-Bell numbers, Fibonacci numbers. \bigskip \hrule \bigskip \noindent (Concerned with sequence \seqnum{A000045}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received August 1, 2003; revised version received January 8, 2004. Published in {\it Journal of Integer Sequences}, February 9 2004. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}. \vskip .1in \end{document}