\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amscd} \usepackage{graphicx} \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} \usepackage{tikz} \usepackage{paralist} \newcommand {\stirlingf}[2]{\genfrac[]{0pt}{}{#1}{#2}} \newcommand {\lah}[2]{\genfrac{\lfloor}{\rfloor}{0pt}{}{#1}{#2}} \newcommand {\stirlings}[2]{\genfrac \{ \}{0pt}{}{#1}{#2}} \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}{\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 The Multivariate Lah and Stirling Numbers } \vskip 1cm \large Amine Belkhir \\ Faculty of Mathematics \\ USTHB \\ BP 32, El Alia \\ 16111 Bab Ezzouar \\ Algiers \\ Algeria \\ \href{mailto:ambelkhir@gmail.com}{\tt ambelkhir@gmail.com} \end{center} \vskip .2 in \begin{abstract} An ordered partition of $\{1,2,\ldots,n\}$ into $k$ blocks $B_{1},B_{2},\ldots,B_{k}$ is a partition where the order of blocks is considered. In the present paper, we we consider the case that each block $B_{i}$ has $r_{i}$ copies. Using this extension of ordered set partitions, we introduce a new generalization of the Lah and Stirling numbers of both kinds which called multivariate Lah and Stirling numbers, respectively. We study several combinatorial properties such as explicit formulas, recurrence relations, generating functions, and some convolution identities. \end{abstract} \section{Introduction} Let $[n] = \{1,\ldots,n\}$. A partition $\pi$ of $[n]$ is a family of nonempty, pairwise disjoint subsets called \emph{blocks}. A partition of $[n]$ into $k$ blocks is denoted $B_{1}/B_{2}/\cdots/B_{k}$ such that $\min(B_{1})<\min(B_{2})<\cdots<\min(B_{k})$. For any $n\geq k\geq 0$, let $\stirlings{n}{k}$, $\stirlingf{n}{k}$ and $\lah{n}{k}$ be the Stirling numbers of the second kind, first kind and Lah numbers, respectively. The numbers $\stirlings{n}{k}$ count the number of set partitions of $[n]$ into $k$ blocks, $\stirlingf{n}{k}$ count the number of partitions of $[n]$ into $k$ cycles. Similarly, the Lah numbers $\lah{n}{k}$ count the number of partitions of set $[n]$ into $k$ nonempty lists. The falling and rising factorials are defined, respectively by \begin{equation*} (x)_{n}=x(x+1)(x+2)\cdots(x+n-1), \ \ \ \ (x)_{0}=1, \end{equation*} and \begin{equation*} \langle x\rangle_{n}= x(x-1)(x-2)\cdots(x-n+1), \ \ \ \ \langle x\rangle_{0}=1. \end{equation*} The Stirling numbers of second kind appear in the expansion $x^{n} = \sum_{k}\stirlings{n}{k} \langle x\rangle_{k}$, and the Stirling numbers of first kind appear in the expansion $(x)_{n}= \sum_{k}\stirlingf{n}{k}x^{k}$. The Lah numbers are connection coefficients between rising and falling factorials $(x)_{n}= \sum_{k}\lah{n}{k}\langle x\rangle_{k}$. The Lah numbers can be expressed in terms of Stirling numbers of second and first kinds \cite[p.\ 156]{Comtet}, as follows: \begin{equation*} \lah{n}{k}=\sum_{j=k}^{n}\stirlingf{n}{j}\stirlings{j}{k}. \end{equation*} The Lah numbers have the following explicit formula \cite[p.\ 134]{Comtet}: \begin{equation*} \lah{n}{k}=\frac{n!}{k!} {n-1\choose k-1}. \end{equation*} An ordered partition $\pi$ of $[n]$ into $k$ blocks is a partition where the order of blocks is important $\pi_{\sigma}=B_{\sigma(1)}/B_{\sigma(2)}/\cdots/B_{\sigma(k)}$, where $\sigma$ is a permutation of $[k]$. The number of ordered set partitions of $[n]$ into $k$ blocks is given by $k!\stirlings{n}{k}$, \cite[p.\ 106]{Mariconda}, as follows: \begin{equation} k!\stirlings{n}{k}=\sum_{{\underset{r_{i}\geq1}{r_{1}+r_{2}+\cdots+r_{k}=n}}}{n\choose r_{1},r_{2},\ldots,r_{k}}, \end{equation} where ${n\choose r_{1},r_{2},\ldots,r_{k}}=\frac{n!}{r_{1}!r_{2}!\cdots r_{k}!}$ is the multinomial coefficient. The coefficients ${n\choose r_{1},r_{2},\ldots,r_{k}}$ have the following horizontal generating function \begin{equation} (x_{1}+x_{2}+\cdots+x_{k})^{n}=\sum_{r_{1}+r_{2}+\cdots+r_{k}=n}{n \choose r_{1},r_{2},\ldots,r_{k}}x_{1}^{r_{1}}x_{2}^{r_{2}}\cdots x_{k}^{r_{k}}. \label{multinomial} \end{equation} The relation \eqref{multinomial} can be generalized to multivariate falling and rising factorials \cite[p.\ 149]{Harris}: \begin{equation} \langle x_{1}+x_{2}+\cdots+x_{k}\rangle_{n}=\sum_{r_{1}+r_{2}+\cdots+r_{k}=n}{n\choose r_{1},r_{2},\ldots,r_{k}}\langle x_{1}\rangle_{r_{1}}\langle x_{2}\rangle_{r_{2}}\cdots \langle x_{k}\rangle_{r_{k}}, \end{equation} and \begin{equation} (x_{1}+x_{2}+\cdots+x_{k})_{n}=\sum_{r_{1}+r_{2}+\cdots+r_{k}=n}{n\choose r_{1},r_{2},\ldots,r_{k}}(x_{1})_{r_{1}}(x_{2})_{r_{2}}\cdots (x_{k})_{r_{k}}. \end{equation} Many authors have investigated the Stirling and Lah numbers; see, for instance, \cite{Bel, BBel,Cheon,toufik,Nyul}. The present paper is organized as follows: Section \ref{sec2} introduces the multivariate Lah numbers \emph{multipartitions} set which generalize the set partitions and ordered set partitions. Section \ref{sec3} presents several properties of the multivariate Lah numbers by algebraic and combinatorial arguments. In the last section, we define the multivariate Stirling numbers of the first and second kinds and we provide an expression for the multivariate Lah numbers in terms of the multivariate Stirling numbers and multinomial Stirling numbers were introduced by Moak \cite{Moak}. \section{Combinatorial definition of multivariate Lah numbers}\label{sec2} Let $\textbf{r}_{k}:=(r_{1},r_{2},\ldots,r_{k})$ be a sequence of nonnegative integers. Now suppose that we have $k$ categories of lists $(C_{1},C_{2},\ldots,C_{k})$ such that $|C_{i}|=r_{i}$. Let $\mathcal{OP}_{n}^{\textbf{r}_{k}}$ be the set partitions of $[n]$ into $(r_{1}+r_{2}+\cdots+r_{k})$-lists. A partition $\pi\in \mathcal{OP}_{n}^{\textbf{r}_{k}}$ is of the form $\pi=B_{1}^{r_{1}}/B_{2}^{r_{2}}/\cdots /B_{k}^{r_{k}}$ where $B_{i}^{r_{i}}=\underset{r_{i} \ {\rm times}}{\underbrace{B_{i}/B_{i}/\cdots /B_{i}}}$. \begin{definition}Let $\pi=B_{1}^{r_{1}}/B_{2}^{r_{2}}/\cdots/B_{k}^{r_{k}}$ be a partition of the set $\mathcal{OP}_{n}^{\textbf{r}_{k}}$. A \emph{multipartition} is a permutation of the multiset $\{B_{1}^{r_{1}},B_{2}^{r_{2}},\ldots, B_{k}^{r_{k}}\}$. We let $\mathfrak{S}_{n,\textbf{r}_{k}}(\pi)$ denote the set of all multipartitions of the multiset $\{B_{1}^{r_{1}},B_{2}^{r_{2}},\ldots, B_{k}^{r_{k}}\}$. \end{definition} \begin{example}Let $\pi=\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{1,2}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{3}}}\textcolor[rgb]{1.00,1.00,1.00}{,}/\underset{B_{2}}{\underline{\textcolor[rgb]{0.00,1.00,0.00}{4,5}}}/\underset{B_{3}}{\underline{\textcolor[rgb]{0.00,0.00,1.00}{6}}}\textcolor[rgb]{1.00,1.00,1.00}{,}$ be a partition of the set $[6]$ into $(2,1,1)$-lists. The set $\mathfrak{S}_{6,\textbf{r}_{3}}(\pi)$ of multipartitions associated with the partition $\pi$ is \begin{gather*} \underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{1,2}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{3}}}\textcolor[rgb]{1.00,1.00,1.00}{,}/\underset{B_{2}}{\underline{\textcolor[rgb]{0.00,1.00,0.00}{4,5}}}/\underset{B_{3}}{\underline{\textcolor[rgb]{0.00,0.00,1.00}{6}}} \textcolor[rgb]{1.00,1.00,1.00}{,} \ \ \ \underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{1,2}}}/\underset{B_{2}}{\underline{\textcolor[rgb]{0.00,1.00,0.00}{4,5}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{3}}}/\underset{B_{3}}{\underline{\textcolor[rgb]{0.00,0.00,1.00}{6}}}; \ \ \ \ \underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{1,2}}}/\underset{B_{2}}{\underline{\textcolor[rgb]{0.00,1.00,0.00}{4,5}}}/\underset{B_{3}}{\underline{\textcolor[rgb]{0.00,0.00,1.00}{6}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{3}}}; \\ \underset{B_{2}}{\underline{\textcolor[rgb]{0.00,1.00,0.00}{4,5}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{1,2}}}/\underset{B_{3}}{\underline{\textcolor[rgb]{0.00,0.00,1.00}{6}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{3}}}; \ \ \ \ \underset{B_{2}}{\underline{\textcolor[rgb]{0.00,1.00,0.00}{4,5}}}/\underset{B_{3}}{\underline{\textcolor[rgb]{0.00,0.00,1.00}{6}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{1,2}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{3}}}; \ \ \ \ \underset{B_{2}}{\underline{\textcolor[rgb]{0.00,1.00,0.00}{4,5}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{1,2}}}/\underset{B_{1}}{\underline{\textcolor[rgb]{1.00,0.00,0.00}{3}}}/\underset{B_{3}}{\underline{\textcolor[rgb]{0.00,0.00,1.00}{6}}}. \end{gather*} \end{example} \begin{definition}For any $n$, $r_{1},r_{2},\ldots,r_{k} \geq1$, the multivariate Lah number, which we denote by $\lah{n}{r_{1},r_{2},\ldots,r_{k}}$, is the number of multipartitions of the set $[n]$ into nonempty $\textbf{r}_{k}$-lists. \begin{equation} \lah{n}{r_{1},r_{2},\ldots,r_{k}}=\sum_{\pi\in \mathcal{OP}_{n}^{\textbf{r}_{k}}}|\mathfrak{S}_{n,\textbf{r}_{k}}(\pi)|. \end{equation} \end{definition} \begin{example}There are $3$ multipartitions of the set $[3]$ into $(1,2)$-lists \begin{gather*} \textcolor[rgb]{1.00,0.00,0.00}{1}/\textcolor[rgb]{0.00,1.00,0.00}{2}/\textcolor[rgb]{0.00,1.00,0.00}{3} \ ; \ \ \ \ \textcolor[rgb]{0.00,1.00,0.00}{2}/\textcolor[rgb]{1.00,0.00,0.00}{1}/\textcolor[rgb]{0.00,1.00,0.00}{3} \ ; \ \ \ \ \textcolor[rgb]{0.00,1.00,0.00}{ 2}/\textcolor[rgb]{0.00,1.00,0.00}{3}/\textcolor[rgb]{1.00,0.00,0.00}{1}. \end{gather*} We have $\lah{3}{1,1}=12$, so the corresponding multipartitions are \begin{gather*} \textcolor[rgb]{1.00,0.00,0.00}{1}/\textcolor[rgb]{0.00,0.00,1.00}{2,3} \ ; \ \ \ \ \textcolor[rgb]{0.00,0.00,1.00}{2,3}/\textcolor[rgb]{1.00,0.00,0.00}{1} \ ; \ \ \ \ \textcolor[rgb]{1.00,0.00,0.00}{1}/\textcolor[rgb]{0.00,0.00,1.00}{3,2} \ ; \ \ \ \ \textcolor[rgb]{0.00,0.00,1.00}{3,2}/ \textcolor[rgb]{1.00,0.00,0.00}{1} \ ; \ \ \ \ \textcolor[rgb]{1.00,0.00,0.00}{1,2}/\textcolor[rgb]{0.00,0.00,1.00}{3} \ ; \ \ \ \ \textcolor[rgb]{0.00,0.00,1.00}{3}/\textcolor[rgb]{1.00,0.00,0.00}{1,2} \ ; \ \ \ \ \\ \textcolor[rgb]{1.00,0.00,0.00}{2,1}/\textcolor[rgb]{0.00,0.00,1.00}{3} \ ; \ \ \ \ \textcolor[rgb]{0.00,0.00,1.00}{3}/ \textcolor[rgb]{1.00,0.00,0.00}{2,1} \ ; \ \ \ \ \textcolor[rgb]{1.00,0.00,0.00}{1,3}/\textcolor[rgb]{0.00,0.00,1.00}{2} \ ; \ \ \ \ \textcolor[rgb]{0.00,0.00,1.00}{2}/\textcolor[rgb]{1.00,0.00,0.00}{1,3} \ ; \ \ \ \ \textcolor[rgb]{1.00,0.00,0.00}{3,1}/\textcolor[rgb]{0.00,0.00,1.00}{2} \ ; \ \ \ \ \textcolor[rgb]{0.00,0.00,1.00}{2}/ \textcolor[rgb]{1.00,0.00,0.00}{3,1}. \end{gather*} \end{example} In the following theorem we provide an explicit formula for the multivariate Lah numbers. \begin{theorem}\label{theo1}For any $n$, $r_{1},r_{2},\ldots,r_{k} \geq1$, we have \begin{equation} \lah{n}{r_{1},r_{2},\ldots,r_{k}}=\frac{(n-1)!}{(r_{1}+r_{2}+\cdots+r_{k}-1)!}{n\choose r_{1},r_{2},\ldots,r_{k},n-\sum_{j=1}^{k}r_{j}}. \label{expklah} \end{equation} \end{theorem} \begin{proof} To construct a multipartition of $[n]$ into $\textbf{r}_{k}$-lists we can do the following. First, we select $r_{1}$ elements from $[n]$, each element corresponding to the beginning of one list of category $C_{1}$. There are ${n\choose r_{1}}$ possibilities. Then, we choose $r_{2}$ elements from the remaining $n-r_{1}$ elements, which we place at the beginning of the lists of category $C_{2}$ with ${n-r_{1}\choose r_{2}}$ possible ways, and so on. We choose $r_{k}$ of the remaining $n-r_{1}-\cdots -r_{k-1}$ elements, which we place at the start of the lists of category $C_{k}$. There are ${n-r_{1}-\cdots -r_{k-1}\choose r_{k}}$ possibilities. The remaining $n-r_{1}-\cdots -r_{k}$ elements can be added with $(r_{1}+\cdots+r_{k})(r_{1}+\cdots+r_{k}+1)\cdots (n-1)$ possibilities. This gives us, \begin{eqnarray*} {n\choose r_{1}}{n-r_{1}\choose r_{2}}&\cdots& {n-r_{1}-\cdots -r_{k-1}\choose r_{k}} \prod_{i=0}^{n-1-r_{1}-\cdots-r_{k})}(n-1-i) \\ &=& \frac{(n-1)!}{(r_{1}+r_{2}+\cdots+r_{k}-1)!}{n\choose r_{1},r_{2},\ldots,r_{k},n-\sum_{j=1}^{k}r_{j}}, \end{eqnarray*} which completes the proof. \end{proof} As particular cases of the multivariate Lah numbers, when $k=1$ we obtain the classical Lah numbers and for $r_{1}+r_{2}+\cdots+r_{k}=n$ we get the multinomial coefficient \begin{equation*} \lah{n}{r_{1},r_{2},\ldots,r_{k}}= {n\choose r_{1},r_{2},\ldots,r_{k}}. \end{equation*} Also, when $r_{i}=1$ for all $i\in[k]$ we obtain the ordered Lah numbers, \begin{equation*} \lah{n}{\underset{k-{\rm times}}{\underbrace{1,1,\ldots,1}}}= n!{n-1\choose k-1}. \end{equation*} Let $\sigma(1),\sigma(2),\ldots,\sigma(k)$ be a permutation of $[k]$. Then \begin{equation*} \lah{n}{r_{1},r_{2},\ldots,r_{k}}=\lah{n}{r_{\sigma(1)},r_{\sigma(2)},\ldots,r_{\sigma(k)}}. \end{equation*} From Relation \eqref{expklah}, we deduce an expression for the multivariate Lah numbers in terms of classical Lah numbers. \begin{corollary} \label{expcoro1} For any $n, r_{1},\ldots,r_{k}\geq 1$, we have \begin{equation} \lah{n}{r_{1},r_{2},\ldots,r_{k}}={r_{1}+r_{2}+\cdots+r_{k}\choose r_{1},r_{2},\ldots,r_{k}}\lah{n}{r_{1}+r_{2}+\cdots+r_{k}}. \label{coro1} \end{equation} \end{corollary} \section{Fundamental properties of multivariate Lah numbers}\label{sec3} In this section, we provide some fundamental properties of the multivariate Lah numbers. We start by given the exponential generating function. \begin{theorem}\label{texpogf} The exponential generating function of the multivariate Lah numbers is \begin{equation} \sum_{n\geq 0}\lah{n}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{n}}{n!}= \prod_{i=1}^{k}\frac{1}{r_{i}!}\left(\frac{t}{1-t}\right)^{r_{i}}. \label{expogf} \end{equation} \end{theorem} \begin{proof} From Theorem \ref{theo1} and relation \eqref{coro1}, we have \begin{align*} \sum_{n\geq 0}\lah{n}{r_{1},\ldots,r_{k}} \frac{t^{n}}{n!} &= \sum_{n\geq 0} \frac{(n-1)!}{(r_{1}+\cdots+r_{k}-1)!}{n\choose r_{1},\ldots,r_{k},n-\sum_{j=1}^{k}r_{j}}\frac{t^{n}}{n!} \\ &= \frac{1}{r_{1}!\cdots r_{k}!}\sum_{n\geq 0} {n-1\choose r_{1}+r_{2}+\cdots+r_{k}-1}t^{n} \\ &= \frac{t^{r_{1}+r_{2}+\cdots+r_{k}}}{r_{1}!\cdots r_{k}!}\sum_{n\geq 0} {n-1\choose n-r_{1}-r_{2}-\cdots-r_{k}}t^{n} \\ &= \frac{t^{r_{1}+r_{2}+\cdots+r_{k}}}{r_{1}!\cdots r_{k}!}\sum_{n\geq 0} {n+r_{1}+r_{2}+\cdots+r_{k}-1\choose n}t^{n} \\ &= \frac{t^{r_{1}+r_{2}+\cdots+r_{k}}}{r_{1}!\cdots r_{k}!}\sum_{n\geq 0} {-r_{1}-r_{2}-\cdots-r_{k}\choose n}(-t)^{n} \\ &= \frac{1}{r_{1}!\cdots r_{k}!}\prod_{i=1}^{k}\left(\frac{t}{1-t}\right)^{r_{i}}, \end{align*} which completes the proof. \end{proof} In the following theorem, we give the multivariate exponential generating function for the multivariate Lah numbers. \begin{theorem} We have \begin{equation} \sum_{n\geq 0}\sum_{r_{1}\geq 0}\sum_{r_{2}\geq 0}\cdots \sum_{r_{k}\geq 0}\lah{n}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{n}}{n!}z_{1}^{r_{1}}z_{2}^{r_{2}}\cdots z_{k}^{r_{k}}= \exp\left(\frac{t}{1-t}(z_{1}+z_{2}+\cdots+z_{k})\right). \end{equation} \end{theorem} \begin{proof} The result follows immediately from Theorem \eqref{texpogf}. \end{proof} An expression for the multivariate falling factorial in terms of the classical rising factorial is given by the following theorem. \begin{theorem}\label{Thgenfac} For any $n\geq 1$, we have \begin{equation} (x_{1}+x_{2}+\cdots+x_{k})_{n}=\sum_{r_{1}+r_{2}+\cdots+r_{k}\leq n} \lah{n}{r_{1},r_{2},\ldots,r_{k}}\langle x_{1}\rangle_{r_{1}} \langle x_{2}\rangle_{r_{2}} \cdots \langle x_{k}\rangle_{r_{k}}, \label{hgenfac} \end{equation} and \begin{equation} \langle x_{1}+x_{2}+\cdots+x_{k}\rangle_{n}=\sum_{r_{1}+r_{2}+\cdots+r_{k}\leq n}(-1)^{n-r_{1}-\cdots-r_{k}} \lah{n}{r_{1},r_{2},\ldots,r_{k}}(x_{1})_{r_{1}}(x_{2})_{r_{2}} \cdots (x_{k})_{r_{k}}. \label{hgenfacc} \end{equation} \end{theorem} \begin{proof} We have \begin{align} \left(1+\frac{t}{1-t}\right)^{x_{1}+x_{2}+\cdots+x_{k}} &= \left(1+\frac{t}{1-t}\right)^{x_{1}}\left(1+\frac{t}{1-t}\right)^{x_{2}}\cdots \left(1+\frac{t}{1-t}\right)^{x_{k}} \nonumber \\ &= \left(\sum_{r_{1}\geq0}\left(\frac{t}{1-t}\right)^{r_{1}}\frac{\langle x_{1}\rangle_{r_{1}}}{r_{1}!}\right) \left(\sum_{r_{2}\geq0}\left(\frac{t}{1-t}\right)^{r_{2}}\frac{\langle x_{2}\rangle_{r_{2}}}{r_{2}!}\right)\nonumber \\ & \ \ \ \times \cdots \times \left(\sum_{r_{k}\geq0}\left(\frac{t}{1-t}\right)^{r_{k}}\frac{\langle r_{k}\rangle_{r_{k}}}{r_{k}!}\right)\nonumber \\ &= \sum_{r_{1},r_{2},\ldots,r_{k}\geq0} \prod_{i=1}^{k}\frac{1}{r_{i}!}\left(\frac{t}{1-t}\right)^{r_{i}}\langle x_{1}\rangle_{r_{1}} \langle x_{2}\rangle_{r_{2}}\cdots \langle x_{k}\rangle_{r_{k}} \nonumber \\ &= \sum_{r_{1},r_{2},\ldots,r_{k}} \left(\sum_{n\geq 0}\lah{n}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{n}}{n!}\right)\langle x_{1}\rangle_{r_{1}} \langle x_{2}\rangle_{r_{2}}\cdots \langle x_{k}\rangle_{r_{k}} \nonumber \\ &= \sum_{n\geq 0}\sum_{r_{1},r_{2},\ldots,r_{k}}\lah{n}{r_{1},r_{2},\ldots,r_{k}}\langle x_{1}\rangle_{r_{1}} \langle x_{2}\rangle_{r_{2}}\cdots \langle x_{k}\rangle_{r_{k}}\frac{t^{n}}{n!}. \label{prfgenfacg} \end{align} The left hand side of the first equality is \begin{align} \left(1+\frac{t}{1-t}\right)^{x_{1}+x_{2}+\cdots+x_{k}} &=\left(1-t\right)^{-(x_{1}+x_{2}+\cdots+x_{k})} \nonumber \\ &= \sum_{n\geq0}(-1)^{n}{-x_{1}-x_{2}-\cdots-x_{k}\choose n}t^{n} \nonumber \\ &= \sum_{n\geq0}{x_{1}+x_{2}+\cdots+x_{k}+n-1\choose n}t^{n} \nonumber\\ &= \sum_{n\geq0} (x_{1}+x_{2}+\cdots+x_{k})_{n}\frac{t^{n}}{n!}. \label{prfgenfacd} \end{align} Equating the coefficients of $\frac{t^{n}}{n!}$ in \eqref{prfgenfacg} and \eqref{prfgenfacd} yields \eqref{hgenfac}. Equation \eqref{hgenfacc} follows by substituting $(-x_{i})$ for $i\in[k]$. \end{proof} Using Theorem \ref{Thgenfac}, we get, for example \begin{align*} (x_{1}+x_{2})_{2} &= \langle x_{1}\rangle_{2}+2x_{1}x_{2}+\langle x_{2}\rangle_{2}+ 2(x_{1}+x_{2}), \\ (x_{1}+x_{2})_{3} &= \langle x_{1}\rangle_{3}+3\langle x_{1}\rangle_{2}x_{2}+3x_{1}\langle x_{2}\rangle_{2}+\langle x_{2}\rangle_{3}+6(\langle x_{1}\rangle_{2}+2x_{1}x_{2}+\langle x_{2}\rangle_{2}) \\ & \ \ \ \ \ +6(x_{1}+x_{2}), \\ (x_{1}+x_{2}+x_{3})_{2} &= \langle x_{1}\rangle_{2}+\langle x_{2}\rangle_{2}+\langle x_{3}\rangle_{2}+2(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}) +2(x_{1}+x_{2}+x_{3}). \\ \langle x_{1}+x_{2}\rangle_{2} &= 3(x_{1})_{2}-6x_{1}-6x_{2}+6x_{1}x_{2}+3(x_{2})_{2}. \\ \end{align*} \begin{theorem} For any $n,r_{1},r_{2},\ldots,r_{k}\geq 1$, we have \begin{equation} \sum_{s_{1}+s_{2}+\ldots+s_{k}\leq n}(-1)^{\sum_{i} s_{i}-r_{i}} \lah{n}{s_{1},s_{2},\ldots,s_{k}}\lah{s_{1}}{r_{1}}\lah{s_{2}}{r_{2}}\cdots \lah{s_{k}}{r_{k}}={n\choose r_{1},r_{2},\ldots,r_{k}}. \end{equation} \end{theorem} \begin{proof} From Relation \eqref{hgenfac}, we have \begin{align*} (x_{1}+x_{2}+\cdots+x_{k})_{n} &= \sum_{s_{1}+s_{2}+\cdots+s_{k}\leq n} \lah{n}{s_{1},s_{2},\ldots,s_{k}}\langle x_{1}\rangle_{s_{1}} \langle x_{2}\rangle_{s_{2}} \cdots \langle x_{k}\rangle_{s_{k}} \\ &= \sum_{s_{1}+s_{2}+\cdots+s_{k}\leq n} \lah{n}{s_{1},s_{2},\ldots,s_{k}} \sum_{r_{1},r_{2},\ldots,r_{k}}(-1)^{\sum_{i} s_{i}-r_{i}}\lah{s_{1}}{r_{1}}\lah{s_{2}}{r_{2}}\cdots \lah{s_{k}}{r_{k}} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times(x_{1})_{r_{1}}(x_{2})_{r_{2}}\cdots (x_{k})_{r_{k}}. \end{align*} On other hand, we have \begin{equation} (x_{1}+x_{2}+\cdots+x_{k})_{n}=\sum_{r_{1}+\cdots+r_{k}=n}{n\choose r_{1},r_{2},\ldots,r_{k}}(x)_{r_{1}}(x)_{r_{2}}\cdots (x)_{r_{k}}. \end{equation} Equating the coefficients of $(x)_{r_{1}}(x)_{r_{2}}\cdots (x)_{r_{k}}$ we obtain the result. \end{proof} \begin{corollary} The following identity holds \begin{equation} (x_{1})_{r_{1}}(x_{2})_{r_{2}}\cdots (x_{k})_{r_{k}}=\sum_{s_{1},s_{2},\ldots,s_{k}}\lah{r_{1}}{s_{1}}\lah{r_{2}}{s_{2}}\cdots \lah{r_{k}}{s_{k}}\langle x_{1}\rangle_{s_{1}} \langle x_{2}\rangle_{s_{2}} \cdots \langle x_{k}\rangle_{s_{k}}. \end{equation} \end{corollary} In the next theorem, we give a convolution identity involving the multivariate Lah numbers. \begin{theorem} \label{theo5} We have \begin{equation} \prod_{i=1}^{k}{r_{i}+s_{i}\choose r_{i}} \lah{n}{r_{1}+s_{1},r_{2}+s_{2},\ldots,r_{k}+s_{k}}=\sum_{j=1}^{n}{n\choose j}\lah{j}{r_{1},r_{2},\ldots,r_{k}}\lah{n-j}{s_{1},s_{2},\ldots,s_{k}}. \label{conv1} \end{equation} \end{theorem} \begin{proof} From Theorem \ref{texpogf}, we have \begin{align*} \sum_{n\geq0} \prod_{i=1}^{k}{r_{i}+s_{i}\choose r_{i}} \lah{n}{r_{1}+s_{1},\ldots,r_{k}+s_{k}}\frac{t^{n}}{n!} &= \prod_{i=1}^{k}{r_{i}+s_{i}\choose r_{i}} \frac{1}{(r_{i}+s_{i})!}\left(\frac{t}{1-t}\right)^{(r_{i}+s_{i})} \\ &= \left( \prod_{i=1}^{k}\frac{1}{(r_{i})!} \left(\frac{t}{1-t}\right)^{r_{i}}\right)\left(\prod_{j=1}^{k}\frac{1}{(s_{j})!} \left(\frac{t}{1-t}\right)^{s_{j}} \right)\\ &= \left(\sum_{l\geq 0}\lah{l}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{l}}{l!}\right) \\ & \ \ \ \ \ \ \ \ \ \ \times \left(\sum_{m\geq 0}\lah{m}{s_{1},s_{2},\ldots,s_{k}}\frac{t^{m}}{m!}\right). \end{align*} Equating the coefficients of $\frac{t^{n}}{n!}$ in both sides, we get the desired result. \end{proof} \begin{proof}[Combinatorial proof of Theorem \ref{theo5}.] Let $C=\{1,2,\ldots, k\}$ be a list of $k$ different colors. The left hand side of the identity counts the number of multipartitions of the set $[n]$ into $(r_{1}+s_{1},\ldots,r_{k}+s_{k})$-lists such that the elements of $r_{i}$ lists among $r_{i}+s_{i}$ lists get colour $i$, for all $i\in [k]$. In the right hand side, we start by choosing $j$ elements from $n$ and there are ${n\choose j}$ ways to do. The $j$ elements have to be partitioned into $(r_{1},\ldots,r_{k})$-lists such that the elements of the lists $r_{i}$ get colour $i$ and the remaining $n-j$ elements have to be partitioned into $(s_{1},\ldots,s_{k})$-lists. \end{proof} In the following theorem we give a generalization of the formula \eqref{conv1}. \begin{theorem}We have \begin{equation} \prod_{i=1}^{k}{r_{i}\choose r_{1,i},\ldots,r_{t,i}} \lah{n}{r_{1},r_{2},\ldots,r_{k}} =\sum_{j_{1},\ldots,j_{t}}{n\choose j_{1},\ldots,j_{t}}\prod_{i=1}^{t}\lah{j_{i}}{r_{1,i},r_{2,i},\ldots,r_{t,i}}. \label{conv11} \end{equation} with $r_{1,i}+r_{2,i}+\cdots+r_{t,i}=r_{i}$ for $i\in [t]$. \end{theorem} Now we give some recurrence relations satisfied by the multivariate Lah numbers. \begin{theorem}The multivariate Lah numbers satisfy the following recurrence relations \begin{asparaenum}[(i)] \item Triangular recurrence relation: \begin{align} \lah{n}{r_{1},r_{2},\ldots,r_{k}} &= \sum_{i=1}^{k}\lah{n-1}{r_{1},\ldots,r_{i}-1,\ldots,r_{k}} \nonumber \\ & \ \ \ \ \ \ +(n+r_{1}+\cdots+r_{k}-1)\lah{n-1}{r_{1},r_{2},\ldots,r_{k}}, \label{triangrec} \end{align} with initial terms $\lah{n}{r_{1},r_{2},\ldots,r_{k}}=0$ if $n< r_{1}+r_{2}+\cdots+r_{k}$. \item Horizontal recurrence relation: \begin{equation} \lah{n}{r_{1},r_{2},\ldots,r_{k}}=\sum_{r_{1}+\cdots+r_{k}\leq j\leq n }(r_{1}+\cdots+r_{k}+j)_{n-j}\sum_{i=1}^{k}\lah{j-1}{r_{1},\ldots,r_{i}-1,\ldots,r_{k}}. \label{horizrec} \end{equation} \item Diagonal recurrence relation: \begin{align} \lah{n}{r_{1},r_{2},\ldots,r_{k}} &= \sum_{j=0}^{r_{1}+\cdots+r_{k}}\sum_{s_{1}+\cdots+s_{k}=j}{j \choose s_{1},\ldots,s_{k}} \nonumber \\ & \ \ \ \ \ \ \ \ \ \times (n+r_{1}+\cdots+r_{k}-2j-1)\lah{n-j-1}{r_{1}-s_{1},\ldots,r_{k}-s_{k}}. \label{crossrec} \end{align} \end{asparaenum} \end{theorem} \begin{proof} Let us show \eqref{triangrec}. A multipartition of the set $[n]$ into $(r_{1},r_{2},\ldots,r_{k})$-lists can be obtained from a multipartition of the set $[n-1]$ into $(r_{1},\cdots,r_{i}-1,\cdots,r_{k})$-lists $(1\leq i\leq k)$ to which we add a single list $\{n\}$ of category $C_{i}$, or from a multipartition of the set $[n-1]$ into $(r_{1},r_{2},\ldots,r_{k})$-lists, by adding the element $\{n\}$ before any existing elements or at the end of any list. Then there are $(n+r_{1}+\cdots+r_{k}-1)\lah{n-1}{r_{1},r_{2},\ldots,r_{k}}$ ways. Next, we show \eqref{horizrec}. For a given $r_{1}+\cdots+r_{k} \leq j\leq n$ and $1\leq i\leq k$, let us consider the elements of $[j-1]$ which are not in the same list with the element $\{n\}$. The number of multipartitions of $[j-1]$ into $(r_{1},\ldots,r_{i}-1,\ldots,r_{k})$-lists is $\lah{j-1}{r_{1},\ldots,r_{i}-1,\ldots,r_{k}}$, and there are $(r_{1}+\cdots+r_{k}+j)_{n-j}$ ways to add the remaining elements of $[j,n-1]$ into $(r_{1},r_{2},\ldots,r_{k})$-lists. Summing over all possible $j$ and $i$ gives the result. Finally, we show \eqref{crossrec}. Let $j$ $(0 \leq j \leq r_{1}+\cdots+r_{k})$ be the number of lists which contain exactly one element, then the number of ways to choose a such lists is ${j\choose s_{1},\ldots,s_{k}}$. Now it remains to count the number of multipartitions of $[j+1,n]$ into $(r_{1}-s_{1},\ldots,r_{k}-s_{k})$-lists. So, the number of multipartitions of $[j+1,n-1]$ into $(r_{1}-s_{1},\ldots,r_{k}-s_{k})$-lists is $\lah{n-j-1}{r_{1}-s_{1},\ldots,r_{k}-s_{k}}$ and there are $(n+r_{1}+\cdots+r_{k}-2j-1)$ ways to add the element $\{n\}$ in any list. Summing up yields the desired result. \end{proof} \section{Multivariate Stirling numbers} \begin{definition}\label{Stir12} For any $n$, $r_{1},r_{2},\ldots,r_{k} \geq0$, the {\it multivariate Stirling} numbers of the first kind, denoted $\stirlingf{n}{r_{1},r_{2},\ldots,r_{k}}$, are defined as the numbers of multipartitions of the set $[n]$ into $(r_{1},r_{2},\ldots,r_{k})$-cycles. Analogously, we define the {\it multivariate Stirling} numbers of second kind, denoted $\stirlings{n}{r_{1},r_{2},\ldots,r_{k}}$, as the numbers of multipartitions of $[n]$ into $(r_{1},r_{2},\ldots,r_{k})$-blocks. \end{definition} From Definition \ref{Stir12}, we deduce that the multivariate Stirling numbers of the first kind satisfy the following recurrence relation \begin{equation} \stirlingf{n}{r_{1},r_{2},\ldots,r_{k}}=\sum_{i=1}^{k}\stirlingf{n-1}{r_{1},\ldots,r_{i}-1,\ldots,r_{k}}+(n-1)\stirlingf{n-1}{r_{1},r_{2},\ldots,r_{k}}, \label{recStirlingf} \end{equation} with $\stirlingf{n}{r_{1},r_{2},\ldots,r_{k}}=0$ if $n< r_{1}+r_{2}+\cdots+r_{k}$. The multivariate Stirling numbers of the second kind satisfy the following recurrence relation \begin{equation} \stirlings{n}{r_{1},r_{2},\ldots,r_{k}}=\sum_{i=1}^{k}\stirlings{n-1}{r_{1},\ldots,r_{i}-1,\ldots,r_{k}}+(r_{1}+r_{2}+\cdots+r_{k})\stirlings{n-1}{r_{1},r_{2},\ldots,r_{k}}. \label{recStirlings} \end{equation} with $\stirlings{n}{r_{1},r_{2},\ldots,r_{k}}=0$ if $n< r_{1}+r_{2}+\cdots+r_{k}$. As particular cases, we have \begin{equation*} \stirlingf{n}{\underset{k-{\rm times}}{\underbrace{1,1,\ldots,1}}} = k!\stirlingf{n}{k}, \ \ \ \ \text{for} \ \ n\geq k, \end{equation*} \begin{equation*} \stirlings{n}{\underset{k-{\rm times}}{\underbrace{1,1,\ldots,1}}} = k!\stirlings{n}{k}, \ \ \ \ \text{for} \ \ n\geq k, \end{equation*} and \begin{equation*} \stirlings{n}{r_{1},r_{2},\ldots,r_{k}} = {n\choose r_{1},r_{2},\ldots,r_{k}}, \ \ \ \ \text{for} \ \ r_{1}+r_{2}+\cdots+r_{k}=n. \end{equation*} Next, we give an explicit formula for the multivariate Stirling of the second kind. \begin{theorem} For any $n\geq 0$, we have \begin{equation} \stirlings{n}{r_{1},r_{2},\ldots,r_{k}}=\frac{1}{r_{1}!r_{2}!\cdots r_{k}!}\sum_{j=0}^{r_{1}+\cdots+r_{k}} (-1)^{j}{r_{1}+\cdots+r_{k}\choose j}(r_{1}+\cdots+r_{k}-j)^{n}. \end{equation} \end{theorem} \begin{proof} The result is obtained by applying the inclusion-exclusion principle. \end{proof} \begin{theorem}\label{stirfacto} For any $n\geq 1$, we have \begin{equation} (x_{1}+x_{2}+\cdots+x_{k})^{n}=\sum_{r_{1}+r_{i}+\cdots+r_{k}\leq n} \stirlings{n}{r_{1},r_{2},\ldots,r_{k}} \langle x_{1}\rangle_{r_{1}} \langle x_{2}\rangle_{r_{2}}\cdots \langle x_{k}\rangle_{r_{k}}. \label{orgfs2} \end{equation} and \begin{equation} (x_{1}+x_{2}+\cdots+x_{k})_{n}=\sum_{r_{1}+r_{i}+\cdots+r_{k}\leq n} \stirlingf{n}{r_{1},r_{2},\ldots,r_{k}}x_{1}^{r_{1}}x_{2}^{r_{2}}\cdots x_{k}^{r_{k}}. \end{equation} \end{theorem} \begin{proof} The result is obtained using induction proof. \end{proof} \begin{theorem}\label{expogfs2} The exponential generating function of the multivariate Stirling numbers of the first and second kind \begin{equation} \sum_{n\geq 0}\stirlingf{n}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{n}}{n!}= \prod_{i=1}^{k}\frac{1}{{r_{i}!}}\left(\ln\left( \frac{1}{1-t}\right)\right)^{r_{i}}, \end{equation} and \begin{equation} \sum_{n\geq 0}\stirlings{n}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{n}}{n!}= \prod_{i=1}^{k}\frac{\left(e^{t}-1\right)^{r_{i}}}{r_{i}!}. \end{equation} \end{theorem} From Theorem \ref{expogfs2}, we obtain \begin{equation} \sum_{n\geq 0}\sum_{r_{i}\geq0}\stirlingf{n}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{n}}{n!}z_{1}^{r_{1}}\cdots z_{k}^{r_{k}}=\left(1-t\right)^{-z_{1}-\cdots-z_{k}}, \end{equation} and \begin{equation} \sum_{n\geq 0}\sum_{r_{i}\geq0}\stirlings{n}{r_{1},r_{2},\ldots,r_{k}}\frac{t^{n}}{n!}z_{1}^{r_{1}}\cdots z_{k}^{r_{k}}=\exp\left((e^{t}-1)(z_{1}+\cdots+z_{k})\right). \end{equation} The connection relation between the multivariate Stirling numbers and classical Stirling numbers is \begin{equation} \stirlings{n}{r_{1},r_{2},\ldots,r_{k}}={r_{1}+r_{2}+\cdots+r_{k}\choose r_{1},r_{2},\ldots,r_{k}}\stirlings{n}{r_{1}+r_{2}+\cdots+r_{k}}, \label{coro11} \end{equation} and \begin{equation} \stirlingf{n}{r_{1},r_{2},\ldots,r_{k}}={r_{1}+r_{2}+\cdots+r_{k}\choose r_{1},r_{2},\ldots,r_{k}}\stirlingf{n}{r_{1}+r_{2}+\cdots+r_{k}}. \label{coro12} \end{equation} In the following theorem we express the multivariate Lah numbers in terms of the multivariate Stirling numbers. \begin{theorem} For any $n\geq 1$, we have \begin{equation} \lah{n}{r_{1},r_{2},\ldots,r_{k}}=\sum_{j_{1}+\cdots+j_{k}=r_{1}+\cdots+r_{k}}^{n} \stirlingf{n}{j_{1},j_{2},\ldots,j_{k}}\stirlings{j_{1}}{r_{1}}\stirlings{j_{2}}{r_{2}}\cdots \stirlings{j_{k}}{r_{k}}. \end{equation} \end{theorem} \begin{proof} The result is obtained from \eqref{hgenfac} and \eqref{orgfs2}. \end{proof} \section{Acknowledgments} The author would like to thank the referee and the editor for helpful comments and suggestions. \begin{thebibliography}{20} \bibitem{Bel} H. 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Zeng, Multinomial convolution polynomials, {\em Discrete Math.} {\bf160} (1996), 219--228. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11B37; Secondary 05A19. \noindent \emph{Keywords: } Lah number, Stirling number, ordered set partition, multinomial coefficient. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A008277}, \seqnum{A132393}, and \seqnum{A271703}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received October 27 2019; revised versions received March 14 2020; April 7 2020. Published in {\it Journal of Integer Sequences}, April 8 2020. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}