\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amscd} \usepackage{graphicx} \usepackage{mathrsfs} \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} \usepackage{latexsym} \usepackage{epsf} \usepackage{breakurl} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\rm \underline{#1}}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\e}{\mathfrak{e}} \newcommand{\frakH}{\mathfrak{H}} \newcommand{\frakO}{\mathfrak{O}} \newcommand{\bbH}{\mathbb{H}} \newcommand{\scC}{\mathscr{C}} \newcommand{\calA}{\mathcal{A}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calD}{\mathcal{D}} \newcommand{\calE}{\mathcal{E}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calG}{\mathcal{G}} \newcommand{\calK}{\mathcal{K}} \newcommand{\calH}{\mathcal{H}} \newcommand{\calI}{\mathcal{I}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calM}{\mathcal{M}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calO}{\mathcal{O}} \newcommand{\calP}{\mathcal{P}} \newcommand{\calQ}{\mathcal{Q}} \newcommand{\calW}{\mathcal{W}} \newcommand{\scB}{\mathscr{B}} \newcommand{\eps}{\epsilon} \newcommand{\veps}{\varepsilon} \newcommand{\vphi}{\varphi} \DeclareMathOperator{\ImNew}{Im} \DeclareMathOperator{\vol}{vol} \renewcommand{\Im}{\ImNew} \DeclareMathOperator{\ReNew}{Re} \renewcommand{\Re}{\ReNew} \DeclareMathOperator{\modNew}{mod} \renewcommand{\mod}{\modNew} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\PSL}{PSL} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Mp}{Mp} \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Li}{Li} \newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\bmat}[1]{\begin{bmatrix}#1\end{bmatrix}} \newcommand{\brmat}[1]{\left\{\begin{matrix}#1\end{matrix}\right\}} \newcommand{\smat}[1]{\bigl[\begin{smallmatrix}#1\end{smallmatrix}\bigr]} \newcommand{\sprod}[1]{\left\langle#1\right\rangle} \newcommand{\ssprod}[1]{\langle#1\rangle} \newcommand{\msprod}[1]{\Big\langle#1\Big\rangle} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\sabs}[1]{\lvert#1\rvert} \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \newcommand{\snorm}[1]{\lVert#1\rVert} \newcommand{\KS}[2]{\bigg(\frac{#1}{#2}\bigg)} \newcommand{\sKS}[2]{(\frac{#1}{#2})} \newcommand{\HypF}[2]{{}_{#1}F_{#2}} \newcommand{\weightkaction}[1]{\,\Big\vert_{#1}\,} \newcommand{\sweightkaction}[1]{\,\big\vert_{#1}\,} \newcommand{\st}{\ \Big\vert\ } \newcommand{\sst}{\ \big\vert\ } \newcommand{\thmref}[2]{\hyperref[#2]{#1 \ref*{#2}}} \renewcommand{\eqref}[1]{\hyperref[#1]{(\ref*{#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 Symmetrized Poly-Bernoulli Numbers and \\ \vskip .1in Combinatorics } \vskip 1cm \large Toshiki Matsusaka\\ Institute for Advanced Research\\ Nagoya University\\ Furo-cho, Chikusa-ku\\ Nagoya 464-8601\\ Japan\\ \href{mailto:matsusaka.toshiki@math.nagoya-u.ac.jp}{\tt matsusaka.toshiki@math.nagoya-u.ac.jp} \end{center} \vskip .2 in \begin{abstract} Poly-Bernoulli numbers are one of the generalizations of the classical Bernoulli numbers. Since a negative indexed poly-Bernoulli number is an integer, it is an interesting problem to study this number from a combinatorial viewpoint. In this short article, we give a new combinatorial relation between symmetrized poly-Bernoulli numbers and Dumont-Foata polynomials. \end{abstract} \section{Introduction} A \textit{poly-Bernoulli polynomial} $B_m^{(\ell)} (x)$ of index $\ell \in \Z$ is defined by the generating series \[ e^{-xt} \frac{\Li_\ell (1 - e^{-t})}{1 - e^{-t}} = \sum_{m=0}^\infty B_m^{(\ell)} (x) \frac{t^m}{m!}, \] where $\Li_\ell (z)$ is the polylogarithm function given by \[ \Li_\ell (z) = \sum_{m=1}^\infty \frac{z^m}{m^\ell} \quad (|z| < 1). \] The polynomial $B_m^{(1)}(x)$ coincides with the classical Bernoulli polynomial $B_m(1-x) = (-1)^m B_m(x)$ since $\Li_1(z) = -\log(1-z)$ holds. Following Kaneko \cite{Kan97}, the special value $B_m^{(\ell)} := B_m^{(\ell)}(0)$ is called a poly-Bernoulli number of index $\ell$. The aim of this study is giving a combinatorial perspective to the special values of $B_m^{(\ell)}(x)$ at integers $k \in \Z$. We assume that the index $\ell \leq 0$. In this case the values $B_m^{(\ell)}(k)$ are always integers. One of the first such investigations was Brewbaker's study \cite{Bre08}. He noticed the coincidence of two numbers, the poly-Bernoulli number $B_m^{(\ell)}(0)$ and the number of $01$ lonesum matrices of size $m \times |\ell|$. Recently, B\'{e}nyi and Hajnal \cite{BH17} also established more combinatorial relations in this direction. In this article, we take a step in another direction similar to Kaneko, Sakurai, and Tsumura \cite{KST18}. To describe this more precisely, let $G_n$ be the \textit{Genocchi number} \seqnum{A110501} defined by $G_n = 2(2^{n+2} - 1)|B_{n+2}|$, where $B_m = B_m^{(1)}(1)$ is the classical Bernoulli number. In \cite[Theorem 4.2]{KST18} the authors showed \begin{align}\label{k=1} \sum_{\ell =0}^n (-1)^\ell B_{n-\ell}^{(-\ell -1)} (1) = (-1)^{n/2} G_n \end{align} for any $n \geq 0$. As they mentioned, this equation is an analogue of the result of Arakawa and Kaneko \cite{AK99}, \begin{align} \label{k=0} \sum_{\ell =0}^n (-1)^\ell B_{n-\ell}^{(-\ell)} (0) = \begin{cases} 1, &\text{if } n = 0;\\ 0, &\text{if } n > 0. \end{cases} \end{align} In addition, Sakurai asked in her master's thesis whether we can generalize these equations for any positive integers $x = k$, and give some combinatorial meaning to them. Our main result provides an answer to these two questions in terms of the Dumont-Foata polynomial as follows. \begin{theorem}\label{main} Let $G_n(x,y,z)$ be the $n$-th Dumont-Foata polynomial defined in $(\ref{DF-poly})$, and $\scB_m^{(-\ell)}(k)$ the symmetrized poly-Bernoulli number defined in $(\ref{SPB})$. Then we have \[ \sum_{\ell=0}^n (-1)^\ell \scB_{n-\ell}^{(-\ell)} (k) = k! \cdot (-1)^{n/2} G_n(1,1,k) \] for any non-negative integers $n, k \geq 0$. In particular, both sides equal zero for odd $n$. \end{theorem} This theorem recovers the equations (\ref{k=1}) and (\ref{k=0}) since $\scB_m^{(-\ell)} (0) = B_m^{(-\ell)} (0), \scB_m^{(-\ell)} (1) = B_m^{(-\ell-1)}(1)$, and $G_n(1,1,1) = G_n$ hold as we see later. \begin{remark} I feel that there are a lot of possibilities of establishing similar identities as this theorem. Recent work of B\'{e}nyi and Hajnal \cite[Section 6]{BH17} pointed out that the sequence of diagonal sums \[ \sum_{\ell = 0}^n B_{n-\ell}^{(-\ell -1)} (1) \quad (n \geq 0) \] appears as \seqnum{A136127} in the {\it On-Line Encyclopedia of Integer Sequences} (OEIS) \cite{Sloane}. Furthermore, there are many types of generalizations of poly-Bernoulli numbers as referred in \cite{CCKJ19}, and analogues such as poly-Euler numbers \cite{OS17}, and poly-cosecant numbers \cite{KPT20}. \end{remark} \section{Definitions} \subsection{Dumont-Foata polynomials} We review the work of Dumont and Foata \cite{DF76} here. For a positive even integer $n \in 2\Z$, we consider the surjective map $p: \{1,2,\dots, n\} \to \{2,4, \dots, n\}$ with $p(x) \geq x$ for each $x \in \{1,2, \dots, n\}$. This map is called a (surjective) \textit{pistol} of size $n$, and corresponds to the following diagram. Here we draw an example for $n = 6$. \begin{figure}[H] \centering \includegraphics[width=50mm]{pic1.eps} \caption{$p(1) = 2, p(2) = p(4) = 4, p(3) = p(5) = p(6) = 6$} \end{figure} Let $\calP_n$ be the set of all pistols of size $n$. For each pistol $p \in \calP_n$, we define three quantities called bulging, fixed, and maximal points. First, the number $x \in \{1, 2, \dots, n\}$ is a \textit{bulging point} of $p \in \calP_n$ if $p(y) < p(x)$ for any $0< y < x$. We let $b(p)$ denote the number of bluging points of $p$. In the diagram, $b(p)$ corresponds to the number of steps of the minimal stair covering all check marks. For the above example $p$, the points $x = 1, 2, 3$ are bulging points, so that $b(p) = 3$. \begin{figure}[H] \centering \includegraphics[width=130mm]{pic2.eps} \caption{minimal stair (left), not minimal stair (right)} \end{figure} Next, the point $x \in \{1, 2, \dots, n\}$ is called a \textit{fixed point} of $p \in \calP_n$ if $p(x) = x$. Finally, the point $x \in \{1, 2, \dots, n-1\}$ is a \textit{maximal point} of $p \in \calP_n$ if $p(x) = n$. We let $f(p)$ and $m(p)$ denote the numbers of fixed points and maximal points of $p \in \calP_n$, respectively. For the above example, we have $f(p) = 2$ and $m(p) = 2$. Dumont and Foata \cite[Th\'eor\`eme\ 1a, 2]{DF76} established the following interesting theorem. \begin{theorem} $\cite{DF76}$ Let $n \in 2\Z_{>0}$ be a positive even integer. The polynomial defined by \begin{align}\label{DF-poly} G_n(x,y,z) := \sum_{p \in \calP_n} x^{b(p)} y^{f(p)} z^{m(p)} \end{align} is a symmetric polynomial in three variables, and satisfies $G_n(1,1,1) = G_n$. \end{theorem} In addition, $G_0(x,y,z) = 1$ and $G_n(x,y,z) = 0$ for a positive odd integer $n \in \Z$. The polynomial $G_n(x,y,z)$ is called the $n$-th \textit{Dumont-Foata polynomial}. Furthermore, Dumont and Foata showed that the polynomial for $n > 0$ has the form $G_n(x,y,z) = xyz F_n(x,y,z)$, and the polynomial $F_n(x,y,z)$ satisfies the recurrence relation \[ F_n(x,y,z) = (x+z)(y+z) F_{n-2}(x,y,z+1) - z^2 F_{n-2}(x,y,z) \] with initial values $F_1(x,y,z) = 0$ and $F_2(x,y,z) = 1$. This implies that the polynomial $G_n(z) := G_n(1,1,z)$ called the \textit{Gandhi polynomial} \seqnum{A036970} satisfies \begin{align}\label{RF} G_{n+2}(z) = z(z+1) G_n(z+1) - z^2 G_n(z) \end{align} with $G_0(z) = 1, G_1(z) = 0$. For instance, there exist three pistols of size $4$. The pistols have $(b(p), f(p), m(p)) = (2,2,1), (2,1,2)$ and $(1,2,2)$, so that the Dumont-Foata polynomial is given by \[ G_4(x,y,z) = x^2 y^2 z + x^2 y z^2 + x y^2 z^2 = xyz(xy + yz + zx). \] Indeed, $G_4(1,1,1) = 3$ coincides the $4$-th Genocchi number given by $G_4 = 2(2^6 -1) |B_6| = 3$. \begin{figure}[H] \centering \includegraphics[width=150mm]{pic3.eps} \caption{All elements of $\calP_4$} \end{figure} \subsection{Symmetrized poly-Bernoulli numbers} Table 1. includes the first few values of poly-Bernoulli numbers $\{B_m^{(-\ell)}(0)\}$ and $\{B_m^{(-\ell)}(1)\}$ with $m, \ell \geq 0$. \begin{table}[H]\label{Table1} \centering \begin{tabular}{|c||c|c|c|c|c|c} \hline $\ell \backslash m$ & 0 & 1 & 2 & 3 &4\\ \hline\hline 0 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 2 & 4 & 8 & 16\\ \hline 2 & 1 & 4 & 14 & 46 & 146\\ \hline 3 & 1 & 8 & 46 & 230 & 1066\\ \hline 4 & 1 & 16 & 146 & 1066 & 6902\\ \hline \end{tabular} \hspace{20mm} \begin{tabular}{|c||c|c|c|c|c|c} \hline $\ell \backslash m$ & 0 & 1 & 2 & 3 &4\\ \hline\hline 0 & 1 & 0 & 0 & 0 & 0\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 2 & 1 & 3 & 7 & 15 & 31\\ \hline 3 & 1 & 7 & 31 & 115 & 391\\ \hline 4 & 1 & 15 & 115 & 675 & 3451\\ \hline \end{tabular} \caption{$B_m^{(-\ell)} (0)$ \seqnum{A099594}, $B_m^{(-\ell)} (1)$ \seqnum{A136126}} \end{table} We can see the symmetric property of these numbers at a glance. On the other hand, for $k \geq 2$ it seems unlikely that such a simple symmetric property can be given. In order to reproduce the symmetric properties for any $k \geq 2$, Kaneko-Sakurai-Tsumura \cite{KST18} considered combinations of $B_m^{(-\ell)} (k)$. To make this precise, let $m, \ell, k \geq 0$ be non-negative integers. We now define the \textit{symmetrized poly-Bernoulli number} $\scB_m^{(-\ell)} (k)$ by \begin{align}\label{SPB} \scB_m^{(-\ell)} (k) = \sum_{j=0}^k \bmat{k \\ j} B_m^{(-\ell - j)} (k), \end{align} where $\bmat{k \\ j}$ is the unsigned Stirling number of the first kind \seqnum{A130534} defined in \cite[Definition 2.5]{AIK14}. This number satisfies the symmetry property \[ \scB_m^{(-\ell)} (k) = \scB_\ell^{(-m)} (k) \] for any $m,\ell, k \geq 0$. Note that \[ \scB_m^{(-\ell)} (0) = B_m^{(-\ell)} (0), \quad \scB_m^{(-\ell)} (1) = B_m^{(-\ell-1)} (1). \] \begin{table}[H] \centering \begin{tabular}{|c||c|c|c|c|c|c} \hline $\ell \backslash m$ & 0 & 1 & 2 & 3 &4\\ \hline\hline 0 & 1 & $-1$ & 1 & $-1$ & 1\\ \hline 1 & 1 & 0 & 0 & 0 & 0\\ \hline 2 & 1 & 2 & 2 & 2 & 2\\ \hline 3 & 1 & 6 & 18 & 42 & 90\\ \hline 4 & 1 & 14 & 86 & 374 & 1382\\ \hline \end{tabular} \hspace{20mm} \begin{tabular}{|c||c|c|c|c|c|c} \hline $\ell \backslash m$ & 0 & 1 & 2 & 3 &4\\ \hline\hline 0 & 2 & 2 & 2 & 2 & 2\\ \hline 1 & 2 & 8 & 20 & 44 & 92\\ \hline 2 & 2 & 20 & 104 & 416 & 1472\\ \hline 3 & 2 & 44 & 416 & 2744 & 15032\\ \hline 4 & 2 & 92 & 1472 & 15032 & 120632\\ \hline \end{tabular} \caption{$B_m^{(-\ell)} (2)$ and $\scB_m^{(-\ell)} (2)$} \end{table} Moreover, the authors showed the following explicit formula for $\scB_m^{(-\ell)} (k)$. \begin{align}\label{EF} \scB_m^{(-\ell)} (k) = \sum_{j=0}^{\min(m, \ell)} j! (k+j)! \left\{ \mat{m + 1 \\ j+1} \right\} \left\{ \mat{\ell + 1 \\ j+1} \right\}, \end{align} where $\brmat{k\\ j}$ is the Stirling number of the second kind \seqnum{A008277} defined in \cite[Definition 2.2]{AIK14}. We prove our main theorem using this formula in the next section. \section{Proof} To prove Theorem \ref{main}, it suffices to show that the function \[ \tilde{G}_n(k) := \frac{(-1)^{n/2}}{k!} \sum_{\ell=0}^n (-1)^\ell \scB_{n-\ell}^{(-\ell)} (k) \] satisfies the recurrence relation (\ref{RF}) for any integer $k \geq 0$. First, we can easily see that $\tilde{G}_0(k) = 1$ and $\tilde{G}_1(k) = 0$, which are the initial cases. Moreover, for any odd integer $n$, $\tilde{G}_n(k) = 0$ follows from the symmetric property of $\scB_m^{(-\ell)}(k)$. For an even integer $n \geq 2$, by the formula (\ref{EF}) we have \begin{align} (-1)^{n/2} &k! \bigg( k(k+1) \tilde{G}_n (k+1) - k^2 \tilde{G}_n (k) - \tilde{G}_{n+2} (k) \bigg)\nonumber\\ &= k \sum_{j=0}^{n/2} j! (k+j+1)! \sum_{\ell = j}^{n - j} (-1)^\ell \left\{ \mat{n-\ell + 1 \\ j+1} \right\} \left\{ \mat{\ell + 1 \\ j+1} \right\} \label{1}\\ &\qquad -k^2 \sum_{j = 0}^{n/2} j! (k+j)! \sum_{\ell = j}^{n-j} (-1)^\ell \left\{ \mat{n-\ell + 1 \\ j+1} \right\} \left\{ \mat{\ell + 1 \\ j+1} \right\} \label{2}\\ &\qquad + \sum_{j=0}^{n/2+1} j! (k+j)! \sum_{\ell = j}^{n+2-j} (-1)^\ell \left\{ \mat{n-\ell +3 \\ j+1} \right\} \left\{ \mat{\ell + 1 \\ j+1} \right\}. \label{3} \end{align} Since $\brmat{k \\ 1} = 1$ holds for any $k \geq 1$, we can split the third line (\ref{3}) according as $j = 0$ or not, which equals \[ k! + \sum_{j=0}^{n/2} (j+1)! (k+j+1)! \sum_{\ell = j +1}^{n+1-j} (-1)^\ell \left\{ \mat{n +3-\ell \\ j+2} \right\} \left\{ \mat{\ell+1 \\ j +2} \right\}. \] Let \begin{align}\label{a} a_{n,j} := \sum_{\ell = j}^{n-j} (-1)^\ell \left\{ \mat{n-\ell + 1 \\ j+1} \right\} \left\{ \mat{\ell + 1 \\ j+1} \right\}. \end{align} Then the total of (\ref{1}), (\ref{2}), and (\ref{3}) equals \begin{align}\label{goal} k! + \sum_{j=0}^{n/2} j!(k+j)! \bigg( k(j+1) a_{n,j} + (j+1)(k+j+1) a_{n+2, j+1} \bigg). \end{align} Once the sum is $0$, the proof completes. By using the generating function given in \cite[Proposition 2.6 (8)]{AIK14}, we have \begin{align*} \frac{t^j}{(1-t)(1-2t) \cdots (1-(j+1)t)} = \sum_{\ell \geq j} \left\{ \mat{\ell+1 \\ j+1} \right\} t^\ell = \sum_{\ell \leq n - j} \left\{ \mat{n-\ell +1 \\ j+1} \right\} t^{n-\ell} \end{align*} for any non-negative integers $n, j \in \Z_{\geq 0}$. Multiplying these two expressions, we obtain \[ \frac{s^j t^j}{(1-s)(1-t) \cdots (1-(j+1)s)(1-(j+1)t)} = \sum_{\ell \geq j} \sum_{k \leq n - j} \left\{ \mat{\ell +1 \\ j +1} \right\} \left\{ \mat{n-k+1 \\ j+1} \right\} s^\ell t^{n-k}. \] By specializing at $s = -x, t = x$, \begin{align}\label{gen} \frac{(-1)^j x^{2j}}{(1-x^2) \cdots (1-(j+1)^2 x^2)} = \sum_{\ell \geq j} \sum_{k \leq n -j} (-1)^\ell \left\{ \mat{\ell +1 \\ j +1} \right\} \left\{ \mat{n-k+1 \\ j+1} \right\} x^{n+\ell -k}. \end{align} Thus, we see that the number $a_{n,j}$ defined in (\ref{a}) appears as the $n$-th coefficient of (\ref{gen}). By the expression of the left-hand side of (\ref{gen}), we easily see that $a_{n,j} = 0$ when $n$ is an odd integer or $2j > n$. Further, we get the initial values $a_{2j, j} = (-1)^j, a_{n,0} = 1$ for even $n$, and the recurrence relation \[ a_{n+2,j} = (j+1)^2 a_{n,j} - a_{n,j-1}. \] Applying this to the equation (\ref{goal}), we get \begin{align*} k! + \sum_{j=0}^{n/2} &j!(k+j)! \bigg( k(j+1) a_{n,j} + (j+1)(k+j+1) ((j+2)^2 a_{n, j+1} - a_{n, j}) \bigg)\\ &= k! + \sum_{j=0}^{n/2} (j+2)(j+2)!(k+j+1)! a_{n,j+1} - \sum_{j=0}^{n/2} (j+1) (j+1)! (k+j)! a_{n,j}. \end{align*} Since $a_{n, n/2+1} = 0$ and $a_{n,0} = 1$, this equals $0$, which concludes the proof of Theorem \ref{main}. \section{Acknowledgments} The author sincerely thanks the referees for helpful comments and suggestions. This work was supported by JSPS KAKENHI Grant Numbers 18J20590 and 20K14292. \begin{thebibliography}{99} \bibitem{AIK14} T. Arakawa, T. Ibukiyama, and M. Kaneko, {\it Bernoulli Numbers and Zeta Functions}, with an appendix by Don Zagier, Springer, 2014. \bibitem{AK99} T. Arakawa and M. Kaneko, On poly-Bernoulli numbers, {\it Comment. Math. Univ. Sanct. Pauli} {\bf 48} (1999), 159--167. \bibitem{BH17} B. B\'{e}nyi and P. Hajnal, Combinatorial properties of poly-Bernoulli relatives, {\it Integers} {\bf 17} (2017), \#A31. \bibitem{Bre08} C. Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, {\it Integers} {\bf 8} (2008), \#A02. \bibitem{CCKJ19} C. B. Corcino, R. B. Corcino, T. Komatsu, and H. Jolany, On multi poly-Bernoulli polynomials, {\it J. Inequal. Spec. Funct.} {\bf 10} (2019), 21--34. \bibitem{DF76} D. Dumont and D. Foata, Une propri\'{e}t\'{e} de sym\'{e}trie des nombres de Genocchi, {\it Bull. Soc. Math. France} {\bf 104} (1976), 433--451. \bibitem{Kan97} M. Kaneko, Poly-Bernoulli numbers, {\it J. Th\'{e}or. Nombres Bordeaux} {\bf 9} (1997), 199--206. \bibitem{KPT20} M. Kaneko, M. Pallewatta, and H. Tsumura, On polycosecant numbers, {\it J. 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Available at \url{https://oeis.org}. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11B68; Secondary 05A15. \noindent \emph{Keywords: } poly-Bernoulli number, Genocchi number, Dumont-Foata polynomial. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A008277}, \seqnum{A036970}, \seqnum{A099594}, \seqnum{A110501}, \seqnum{A130534}, \seqnum{A136126}, and \seqnum{A136127}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received April 2 2020; revised version received July 15 2020. Published in {\it Journal of Integer Sequences}, October 13 2020. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}