<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amscd}
\usepackage{graphicx}
\usepackage{diagbox}

\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}

\DeclareMathOperator{\bl}{bl}

\begin{document}

\newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\rm \underline{#1}}}

\newcommand{\lah}[2]{\genfrac \lfloor \rfloor {0pt} {0} {#1} {#2}} 
\newcommand{\laht}[2]{\genfrac \lfloor \rfloor {0pt} {1} {#1} {#2}} 
\newcommand{\rlah}[3]{{\genfrac \lfloor \rfloor {0pt} {0} {#1} {#2}}_{#3}} 
\newcommand{\rlaht}[3]{{\genfrac \lfloor \rfloor {0pt} {1} {#1} {#2}}_{#3}} 
\newcommand{\lrlah}[4]{{\genfrac \lfloor \rfloor {0pt} {0} {#1} {#2}}^{(#3)}_{#4}} 
\newcommand{\lrlaht}[4]{{\genfrac \lfloor \rfloor {0pt} {1} {#1} {#2}}^{(#3)}_{#4}} 
\newcommand{\stone}[2]{\genfrac [ ] {0pt} {0} {#1} {#2}}
\newcommand{\stonet}[2]{\genfrac [ ] {0pt} {1} {#1} {#2}}
\newcommand{\rstone}[3]{{\genfrac [ ] {0pt} {0} {#1} {#2}}_{#3}}
\newcommand{\rstonet}[3]{{\genfrac [ ] {0pt} {1} {#1} {#2}}_{#3}}
\newcommand{\lrstone}[4]{{\genfrac [ ] {0pt} {0} {#1} {#2}}^{(#3)}_{#4}} 
\newcommand{\lrstonet}[4]{{\genfrac [ ] {0pt} {1} {#1} {#2}}^{(#3)}_{#4}} 
\newcommand{\sttwo}[2]{\genfrac \{ \} {0pt} {0} {#1} {#2}}
\newcommand{\sttwot}[2]{\genfrac \{ \} {0pt} {1} {#1} {#2}}
\newcommand{\rsttwo}[3]{{\genfrac \{ \} {0pt} {0} {#1} {#2}}_{#3}}
\newcommand{\rsttwot}[3]{{\genfrac \{ \} {0pt} {1} {#1} {#2}}_{#3}}
\newcommand{\lrsttwo}[4]{{\genfrac \{ \} {0pt} {0} {#1} {#2}}^{(#3)}_{#4}} 
\newcommand{\lrsttwot}[4]{{\genfrac \{ \} {0pt} {1} {#1} {#2}}^{(#3)}_{#4}} 


\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 $(l,r)$-Lah Numbers
		}
		\vskip 1cm
		\large
		Aleks \v{Z}igon Tankosi\v{c}\\
		Milojka \v{S}trukelj Elementary School\\
		Delpinova ulica 7\\
		5000 Nova Gorica\\
		Slovenia\\
		\href{mailto:zigontaleks@gmail.com}{\tt zigontaleks@gmail.com}
	\end{center}

\vskip .2in
	
	\begin{abstract}
After reviewing the definitions and some properties of the Lah numbers and the Stirling numbers of both kinds, as well as their generalizations ($r$-Lah numbers, $r$-Stirling numbers of both kinds and $(l,r)$-Stirling numbers of both kinds), we define the $(l,r)$-Lah numbers analogously, prove a recurrence relation that they satisfy, express them explicitly as a multiple sum, and present the difference-differential equations satisfied by their column and row generating functions, respectively. Finally, we pose two conjectures, based on experimental evidence.
	\end{abstract}
	
	
	
	\section{Introduction}
	
After binomial coefficients, the \textit{Stirling numbers of the first
and second kind} are probably the best known triangular arrays of
natural numbers, arising in many combinatorial, algebraic, and even
analytic contexts. Recognized at least since the 18th century, they
were joined and nicely complemented by the \textit{Lah numbers} (a.k.a.\
\textit{Stirling numbers of the third kind}) in the 1950s. In the 1980s,
a third parameter was added to the definition of Stirling numbers,
resulting in the \textit{$r$-Stirling numbers of the first and second
kind}. In the last decade, the \textit{$r$-Lah numbers} were defined and
their properties explored. In 2021, a fourth parameter was added to the
definition of $r$-Stirling numbers, yielding the \textit{$(l,r)$-Stirling
numbers of the first and second kind}. Here we define and explore the
analogous \textit{$(l,r)$-Lah numbers}.

	The article is organized as follows. Section~\ref{two} contains
	definitions of the Lah and $r$-Lah numbers, lists some recurrence
	relations that they satisfy, expresses them explicitly in terms
	of factorials, and presents some of their applications. With
	the exception of explicit representation, Section~\ref{three}
	does the same for the Stirling, $r$-Stirling, and $(l,r)$-Stirling
	numbers of both kinds. In Section~\ref{four}, we define the
	new $(l,r)$-Lah numbers, prove a recurrence relation that they
	satisfy, express them explicitly in terms of a multiple sum,  and
	present the difference-differential equations satisfied by their
	column and row generating functions, respectively. Finally,
	in Section~\ref{five}, we pose two conjectures, based on
	experimental evidence.

	\section{The Lah and $r$-Lah numbers} \label{two}

	The \textit{unsigned Lah numbers} $\laht{n}{k}$ count partitions
	of the set $[n] := \{1,2,\dots,n\}$ into $k$ non-empty linearly
	ordered blocks (\textit{lists}, for short).

	For $n, k \ge 1$, they satisfy the recurrence \begin{align}
		\label{LahRec} \lah{n}{k} = (n+k-1) \lah{n-1}{k} +
		\lah{n-1}{k-1}
	\end{align} which, together with the initial conditions
	$\laht{0}{0} = 1$,  $\laht{n}{0} = \laht{0}{k} = 0$ for $n, k &gt;
	0$, can be used to compute $\laht{n}{k}$ for all $n, k \ge 0$.

	\begin{table}[ht]
		\begin{center}
			\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
				\hline \diagbox[width=2em]{$n$}{$k$} &amp;
				0&amp; 1 &amp; 2&amp; 3 &amp;4 &amp; 5 &amp; 6 &amp; 7 &amp;8 \\ \hline
				0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp;  0 &amp; 0 &amp; 0 &amp;0 &amp;0  \\
				\hline 1 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp;
				0 &amp; 0 \\ \hline 2 &amp; 0 &amp; 2 &amp; 1 &amp; 0 &amp; 0 &amp;
				0 &amp; 0 &amp; 0 &amp; 0 \\ \hline 3 &amp; 0 &amp; 6 &amp; 6 &amp;
				1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\ \hline 4 &amp; 0 &amp;
				24 &amp; 36 &amp; 12 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0  \\
				\hline 5 &amp; 0 &amp; 120 &amp; 240 &amp; 120 &amp; 20 &amp;
				1 &amp; 0  &amp; 0 &amp; 0 \\ \hline 6 &amp; 0 &amp;  720  &amp;
				1800 &amp; 1200 &amp; 300 &amp; 30 &amp; 1 &amp; 0 &amp; 0 \\
				\hline 7 &amp; 0 &amp; 5040 &amp; 15120 &amp; 12600 &amp;
				4200 &amp; 630 &amp; 42 &amp; 1 &amp; 0 \\ \hline 8 &amp;
				0 &amp; 40320&amp;  141120 &amp; 141120 &amp; 58800 &amp;
				11760 &amp; 1176 &amp; 56 &amp; 1 \\ \hline
			\end{tabular}
		\end{center} \caption{Unsigned Lah numbers $\laht{n}{k}$
		for $0 \le n, k\le 8$.}
	\end{table}


	They can also be expressed explicitly in terms of factorials
	and binomial coefficients as \begin{align}
		\label{LahExpl} \lah{n}{k} = \frac{n!}{k!} {n-1 \choose
		n-k}.
	\end{align} For simple combinatorial proofs of
	(\ref{LahRec}) and (\ref{LahExpl}), see, e.g.,
	\cite{petkovvsek2007combinatorial}. Another interesting property
	of Lah numbers is that they appear in the coefficients in the
	expansion of rising powers \[ x^{\overline{n\rule{0mm}{1.5mm}}}
	= x (x+1) (x+2) \cdots (x+n-1) \] in terms of falling powers
	\[ x^{\underline{n}} = x(x-1)(x-2) \cdots (x-n+1) \] and
	vice versa, namely: \[ x^{\overline{n\rule{0mm}{1.5mm}}}\
	=\ \sum_{k=0} ^ {n} \lah{n}{k} x^{\underline{k}},\qquad
	x^{\underline{n\rule{0mm}{1.5mm}}}\ =\	 \sum_{k=0} ^ {n}
	(-1)^{n+k}\lah{n}{k} x^{\overline{k} } \] for all $n \ge
	0$ and all $x$, where $x^{\overline{0\rule{0mm}{1.5mm}}} =
	x^{\underline{0\rule{0mm}{1.5mm}}} = 1$. These identities can
	easily be proved by induction on $n$.

	Lah numbers appear not only in combinatorics and algebra, but also
	in other areas of mathematics, such as mathematical analysis. For
	instance, the $n$-th derivative of the function $e^{\frac{1}{x}}$
	can be expressed for all $n \ge 0$ as \begin{align}
		\label{e1/x} \left(e^{\frac{1}{x}}\right)^{(n)}\ =\
		e^{\frac{1}{x}} (-1)^n \sum_{k=0}^n \lah{n}{k} x^{-(n +
		k)}\ =\ e^{\frac{1}{x}} \sum_{k=0}^n \lah{n}{k}' x^{-(n
		+ k)},
	\end{align} which again can be proved by induction on $n$. Here
	$\laht{n}{k}' = (-1)^{n} \laht{n}{k}$ are the \textit{signed Lah
	numbers}, introduced in \cite{lah1954} by Slovenian mathematician,
	statistician, and actuary Ivo Lah (1896--1979), and later named
	after him by Riordan in \cite[pp.~43--44]{riordan1958}.

	In the last decade, the \textit{$r$-Lah numbers}, denoted by
	$\rlaht{n}{k}{r}$ (counting partitions of $[n]$ into $k$
	non-empty lists such that the elements $1, 2, \dots, r$
	are in distinct lists) were defined, and their properties
	explored (see  \cite{cheonjung2012, belbachir2013cross,
	belbachir2014combinatorial, shattuck2014arXiv,
	nyul2015r, shattuck2016, nyul2021r}; note however that in
	\cite{shattuck2014arXiv,nyul2015r,shattuck2016, nyul2021r},
	the $r$-Lah number with parameters $n, k, r$ equals our $r$-Lah
	number with parameters $n+r, k+r, r$).

	For $n, k \ge 1$, the $r$-Lah numbers satisfy the recurrence
	\begin{align}
		\label{rLahRec} \rlah{n}{k}{r} =\ (n+k-1) \rlah{n-1}{k}{r}
		+ \rlah{n-1}{k-1}{r}
	\end{align} which, together with the initial conditions
	$\rlaht{n}{k}{r} \!\!= 0$ if $n&lt;k$ or $k&lt;r$, and $\rlaht{n}{n}{n}
	\!\!= 1$, allows us to compute $\rlaht{n}{k}{r}$ for all $n, k,
	r \ge 0$.  \begin{table}[ht]
		\begin{center}
			\begin{tabular}{|c|c|c|c|c|c|c|c|}
				\hline \diagbox[width=2em]{$n$}{$k$} &amp;
				2&amp; 3 &amp;4 &amp; 5 &amp; 6 &amp; 7 &amp;8 \\ \hline 2 &amp; 1 &amp;
				0 &amp; 0 &amp; 0 &amp;  0 &amp; 0 &amp; 0	\\ \hline 3 &amp; 4 &amp;
				1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0  \\ \hline 4 &amp; 20 &amp;
				10 &amp; 1 &amp; 0 &amp; 0 &amp;  0 &amp; 0  \\ \hline 5 &amp;
				120 &amp; 90 &amp; 18 &amp; 1 &amp; 0 &amp; 0 &amp; 0  \\ \hline
				6 &amp; 840 &amp; 840 &amp; 252 &amp; 28 &amp; 1 &amp; 0 &amp; 0 \\
				\hline 7 &amp; 6720 &amp; 8400 &amp; 3360 &amp; 560 &amp;
				40 &amp; 1 &amp; 0 \\ \hline 8 &amp; 60480 &amp; 90720 &amp;
				45360 &amp; 10080 &amp; 1080 &amp; 54 &amp; 1 \\ \hline
			\end{tabular}
		\end{center} \caption{$2$-Lah numbers $\rlaht{n}{k}{2}$
		for $2 \le n, k\le 8$.}
	\end{table} They are expressed explicitly by the formula
	\begin{align}
		\label{rLahExpl} \rlah{n}{k}{r} =\
		\frac{(n+r-1)!}{(k+r-1)!} {n-r \choose k-r}.
	\end{align}

	Note that $r$-Lah numbers have applications in graph theory
	\cite{nyul2021r}. They also satisfy identities connecting rising
	and falling powers such as \begin{align}
		(x + 2r)^{\overline{n}}\ &amp;=\ \sum_{k=0}^ {n}
		\rlah{n+r}{k+r}{r} x^{\underline{k}},\label{rfp1}\\
		(x - r)^{\underline{n}}\ &amp;=\ \sum_{k=0}^ {n} (-1) ^
		{n+k} \rlah{n+r}{k+r}{r} (x+r)^{\overline{k}}.\label{rfp2}
	\end{align} For proofs of identities
	(\ref{rLahRec})--(\ref{rfp2}), see \cite{belbachir2013cross}
	and \cite{nyul2015r}.




	\section{The Stirling, $r$-Stirling, and $(l,r)$-Stirling numbers}
	\label{three}

	\textit{Stirling numbers} were introduced by the Scottish
	mathematician James Stirling (1692--1770) in 1730.  They come
	in two flavors: unsigned Stirling numbers of the first kind
	$\stonet{n}{k}$ count permutations of the set $[n]$ with exactly
	$k$ cycles (or, equivalently, partitions of the set $[n]$ into
	$k$ cyclically ordered blocks), while unsigned Stirling numbers
	of the second kind $\sttwot{n}{k}$ count partitions of the set
	$[n]$ into $k$ unordered blocks.

	\textit{Unsigned Stirling numbers of the first kind} for $n,
	k \ge 1$ satisfy the recurrence \begin{align}
		\label{St1Rec} \stone{n}{k} = (n-1) \stone{n-1}{k}
		+ \stone{n-1}{k-1}
	\end{align} which, together with the initial conditions
	$\stonet{0}{0} = 1$,  $\stonet{n}{0} = \stonet{0}{k} = 0$ for
	$n, k &gt; 0$, can be used to compute $\stonet{n}{k}$ for all $n,
	k \ge 0$.  They are also the coefficients in the expansion of
	rising powers in terms of ordinary powers: \begin{align}
		\label{St1Powers} x^{\overline{n}}\ =\ \sum_{k=0}^n
		\stone{n}{k} x^k.
	\end{align}

	Similarly, \textit{unsigned Stirling numbers of the second kind}
	for $n, k \ge 1$ satisfy the recurrence \begin{align}
		\label{St2Rec} \sttwo{n}{k} = k \sttwo{n-1}{k} +
		\sttwo{n-1}{k-1}
	\end{align} which, together with the initial conditions
	$\sttwot{0}{0} = 1$,  $\sttwot{n}{0} = \sttwot{0}{k} = 0$ for
	$n, k &gt; 0$, can be used to compute $\sttwot{n}{k}$ for all $n,
	k \ge 0$.  They are also the coefficients in the expansion of
	ordinary powers in terms of falling powers: \begin{align}
		\label{St2Powers} x^n\ =\ \sum_{k=0}^n	\sttwo{n}{k}
		x^{\underline{k}}.
	\end{align} There is a nice identity expressing Lah numbers with
	Stirling numbers of both kinds, namely \begin{align}
		\label{lahst12} \lah{n}{k}\ =\ \sum_{j=0}^{n} \stone{n}{j}
		\sttwo{j}{k}.
	\end{align} For proofs of identities
	(\ref{St1Rec})--(\ref{lahst12}), see, e.g.,  \cite{GKP89}  or
	\cite{petkovvsek2007combinatorial}.

	In the 1980's, Stirling numbers of both kinds were generalized to
	$r$-Stirling numbers (cf.\ \cite{carlitz1980_1, carlitz1980_2,
	broder1984r}). The \textit{unsigned $r$-Stirling numbers of the
	first kind} $\rstonet{n}{k}{r}$ count permutations of $[n]$
	having $k$ cycles such that the numbers $1,2,\dots,r$ are in
	distinct cycles.  They satisfy the same recurrence as the numbers
	$\stonet{n}{k}$ \begin{align}
		\rstone{n}{k}{r} &amp;=\ (n-1) \rstone{n-1}{k}{r} +\
		\rstone{n-1}{k-1}{r}, \quad \text{for }\ n, k \ge
		1,\label{rec1rstone}
	\end{align} which, together with initial conditions
	$\rstonet{n}{k}{r} = 0$ for $n &lt; k$ or $k &lt; r$, and
	$\rstonet{n}{n}{r} = 1$ for $r \le n$, can be used to compute
	$\rstonet{n}{k}{r}$ for all $n,k,r \ge 0$. Another, so-called
	`cross' recurrence (in which $r$ changes but $n$ stays fixed)
	satisfied by the unsigned $r$-Stirling numbers of the first kind
	is \begin{align}
		\rstone{n}{k}{r} &amp;=\ \frac{1}{r-1}
		\left(\rstone{n}{k-1}{r-1} -\ \rstone{n}{k-1}{r}\right),
		\quad \text{for }\ n, k \ge 1,\ r \ge 2 \label{rec2rstone}
	\end{align} with initial conditions $\rstonet{n}{k}{r}
	= \stonet{n}{k}$ for $r \le 1$, and $\rstonet{n}{k}{r} = 0$
	for $n &lt; k$ or $k &lt; r$.

	The \textit{unsigned $r$-Stirling numbers of the second kind}
	$\rsttwot{n}{k}{r}$ count partitions of $[n]$ into $k$ nonempty
	disjoint subsets such that the numbers $1,2,\dots,r$ are in
	distinct subsets. They satisfy the same recurrence as the numbers
	$\sttwot{n}{k}$ \begin{align}
		\rsttwo{n}{k}{r} &amp;=\ k \rsttwo{n-1}{k}{r} +\
		\rsttwo{n-1}{k-1}{r}, \quad \text{for }\ n, k \ge
		1,\label{rec1rsttwo}
	\end{align} which, together with initial conditions
	$\rsttwot{n}{k}{r} = 0$ for $n &lt; k$ or $k &lt; r$, and
	$\rsttwot{n}{n}{r} = 1$ for $r \le n$, can be used to compute
	$\rsttwot{n}{k}{r}$ for all $n,k,r \ge 0$. The `cross' recurrence
	satisfied by $\rsttwot{n}{k}{r}$ is \begin{align}
		\rsttwo{n}{k}{r} &amp;=\ \rsttwo{n}{k}{r-1} -\
		(r-1)\rsttwo{n-1}{k}{r-1}, \quad \text{for }\ n, k,
		r \ge 1 \label{rec2rsttwo}
	\end{align} with initial conditions $\rsttwot{n}{k}{0} =
	\sttwot{n}{k}$, and $\rsttwot{n}{k}{r} = 0$ for $n &lt; k$ or $k
	&lt; r$.

	For proofs of identities (\ref{rec1rstone})--(\ref{rec2rsttwo}),
	see   \cite{broder1984r}.

	In 2021, Belbachir and Djemmada \cite{belbachir2021r}
	introduced \textit{$(l,r)$-Stirling numbers of the first kind}
	$\lrstonet{n}{k}{l}{r}$ and \textit{$(l,r)$-Stirling numbers of
	the second kind} $\lrsttwot{n}{k}{l}{r}$ which count ordered
	$l$-tuples $(\pi_1, \pi_2, \dots, \pi_l)$ of partitions of
	$[n]$ into $k$ cyclically ordered blocks, respectively into $k$
	unordered blocks, such that the numbers $1, 2, \dots, r$ are
	in distinct cycles, and satisfy \[ \bl \pi_{1} = \bl \pi_{2} =
	\dots = \bl \pi_{l} \] where for $i=1,2,\dots,l$ and $\pi_i =
	\{b_1^{(i)}, b_2^{(i)}, \dots, b_k^{(i)}\}$, where $b_1^{(i)},
	\ldots b_k^{(i)}$ are the blocks of partition $\pi_i$, we denote
	by \[ \bl \pi_i = \{\min b_1^{(i)}, \min b_2^{(i)}, \dots,
	\min b_k^{(i)}\} \] the set of \textit{block leaders}, i.e.,
	of \textit{minima} of the blocks of partition $\pi_i$.

	Note that the $(l,r)$-Stirling numbers of the second kind are a
	generalization of the \textit{central factorial numbers} of the
	second kind, see, e.g., \cite{kim2022}.



	\begin{example}
		Let us compute $\lrsttwot{4}{3}{2}{2}$. Here $n=4$,
		$k=3$, $l=r=2$, so we need to construct all partitions
		of the set $[4]=\{1,2,3,4\}$ into three nonempty blocks,
		such that 1 and 2 are in distinct blocks. There are five
		such partitions: \begin{align*}
			\pi_1 &amp;= \{\{1\},\{2\},\{3,4\}\},\\ \pi_2
			&amp;= \{\{1\},\{2,3\},\{4\}\},\\ \pi_3 &amp;=
			\{\{1\},\{2,4\},\{3\}\},\\ \pi_4 &amp;=
			\{\{1,3\},\{2\},\{4\}\},\\ \pi_5 &amp;=
			\{\{1,4\},\{2\},\{3\}\},
		\end{align*} and the sets of their block leaders are
		\begin{align*}
			\bl \pi_1 &amp;= \bl \pi_3 = \bl \pi_5 = \{1,2,3\},\\
			\bl \pi_2 &amp;= \bl \pi_4 = \{1,2,4\}.
		\end{align*} Now we compute the number of ordered
		$l$-tuples (i.e., ordered pairs) of partitions
		$\pi_1,\pi_2,\pi_3,\pi_4,\pi_5$ such that partitions in
		the same pair share the same set of block leaders. As
		there are three partitions with the set of block leaders
		equal to $\{1,2,3\}$,  and two partitions with the set
		of block leaders equal to $\{1,2,4\}$, we find that \[
		\lrsttwo{4}{3}{2}{2} =\ 3^2 + 2^2\ = 13.  \]
	\end{example}


	The $(l,r)$-Stirling numbers of the first kind
	$\lrstonet{n}{k}{l}{r}$ satisfy the recurrence


	\begin{align}
		\lrstone{n}{k}{l}{r} &amp;=\ (n-1)^l \lrstone{n-1}{k}{l}{r}
		+ \lrstone{n-1}{k-1}{l}{r}, \quad \text{for }\ n &gt;
		k \ge r \ge 0,\label{rec1lrstone}
	\end{align} which, together with initial conditions
	$\lrstonet{n}{k}{l}{r} = 0$ for $n &lt; k$ or $k &lt; r$, and
	$\lrstonet{n}{k}{l}{n} = \delta_{n,k}$, where $\delta_{n,k} =
	1$ if $n=k$ and $\delta_{n,k} = 0$ if $n \neq k$, can be used
	to compute the numbers $\lrstonet{n}{k}{l}{r}$ for all $n,k,l,r
	\ge 0$. They also satisfy the `cross' recurrence \begin{align}
		\lrstone{n}{k}{l}{r} &amp;=\ \frac{1}{(r-1)^l}
		\left(\lrstone{n}{k-1}{l}{r-1} -\
		\lrstone{n}{k-1}{l}{r}\right), \quad \text{for }\ n, k,
		l \ge 1,\ r \ge 2. \label{rec2lrstone}
	\end{align}

	The $(l,r)$-Stirling numbers of the second kind
	$\lrsttwot{n}{k}{l}{r}$ satisfy the recurrence \begin{align}
		\lrsttwo{n}{k}{l}{r} &amp;=\ k^l \lrsttwo{n-1}{k}{l}{r} +
		\lrsttwo{n-1}{k-1}{l}{r}, \quad \text{for }\ n &gt; k \ge
		r \ge 0,\label{rec1lrsttwo}
	\end{align} which, together with initial conditions
	$\lrsttwot{n}{k}{l}{r} = 0$ for $n &lt; k$ or $k &lt; r$, and
	$\lrsttwot{n}{k}{l}{n} = \delta_{n,k}$, can be used to compute
	the numbers $\lrsttwot{n}{k}{l}{r}$ for all $n,k,l,r \ge 0$. They
	also satisfy the `cross' recurrence \begin{align}
		\lrsttwo{n}{k}{l}{r} &amp;=\ \lrsttwo{n}{k}{l}{r-1} -\
		(r-1)^l\lrsttwo{n-1}{k}{l}{r-1}, \quad \text{for }\ n,
		k, l, r \ge 1. \label{rec2lrsttwo}
	\end{align} For proofs of identities
	(\ref{rec1lrstone})--(\ref{rec2lrsttwo}), see
	\cite{belbachir2021r}.



	\section{The $(l,r)$-Lah numbers} \label{four}

	\begin{definition}
		\label{lrLah_def} \emph{$(l,r)$-Lah numbers}
		$\lrlaht{n}{k}{l}{r}$ count ordered $l$-tuples $(\pi_1,
		\pi_2, \dots, \pi_l)$ of partitions of $[n]$ into $k$
		linearly ordered blocks (lists, for short) such that the
		numbers $1, 2, \dots, r$ are in distinct lists, and \[ \bl
		\pi_{1} = \bl \pi_{2} = \dots = \bl \pi_{l} \] where for
		$i = 1,2,\ldots,l$ and $\pi_i = \{b_1^{(i)}, b_2^{(i)},
		\ldots, b_k^{(i)}\}$, where $b_1^{(i)}, b_2^{(i)},
		\ldots, b_k^{(i)}$ are the blocks of partition $\pi_i$,
		\[ \bl \pi_i = \{\min b_1^{(i)}, \min b_2^{(i)}, \dots,
		\min b_k^{(i)}\} \] is the set of block leaders, i.e.,
		of minima of the lists in partition $\pi_i$.
	\end{definition}

	\begin{example}
		Let us compute $\lrlaht{4}{3}{2}{2}$. Here $n=4$, $k=3$,
		$l=r=2$, so we need to construct all partitions of the
		set $[4]=\{1,2,3,4\}$ into three nonempty disjoint lists,
		such that 1 and 2 are in distinct lists. There are 10
		such partitions: \begin{align*}
			\pi_1 &amp;= \{(1),(2),(3,4)\},\\ \pi_2 &amp;=
			\{(1),(2),(4,3)\},\\ \pi_3 &amp;= \{(1),(2,3),(4)\},\\
			\pi_4 &amp;= \{(1),(3,2),(4)\},\\ \pi_5 &amp;=
			\{(1),(2,4),(3)\},\\ \pi_6 &amp;= \{(1),(4,2),(3)\},\\
			\pi_7 &amp;= \{(1,3),(2),(4)\},\\ \pi_8 &amp;=
			\{(3,1),(2),(4)\},\\ \pi_9 &amp;= \{(1,4),(2),(3)\},\\
			\pi_{10} &amp;= \{(4,1),(2),(3)\},
		\end{align*} and the sets of their block leaders are
		\begin{align*}
			\bl \pi_1 &amp;= \bl \pi_2 = \bl \pi_5 = \bl \pi_6
			= \bl \pi_9 = \bl \pi_{10} = \{1,2,3\},\\ \bl
			\pi_3 &amp;= \bl \pi_4 =  \bl \pi_7 =  \bl \pi_8 =
			\{1,2,4\}.
		\end{align*} Now we compute the number of ordered
		$l$-tuples (i.e., ordered pairs) of partitions
		$\pi_1,\pi_2,\dots,\pi_{10}$ such that partitions in
		the same pair share the same set of block leaders. As
		there are six partitions with the set of block leaders
		equal to $\{1,2,3\}$,  and four partitions with the set
		of block leaders equal to $\{1,2,4\}$, we find that \[
		\lrlah{4}{3}{2}{2} =\ 6^2 + 4^2\ =\ 52.  \]
	\end{example}



	\begin{theorem}
		For $n \ge k \ge r \ge 1$, $(l,r)$-Lah numbers satisfy
		the recurrence relation \begin{equation}
			\label{rec1} \lrlah{n}{k}{l}{r} =  (n+k-1)^l
			\lrlah{n-1}{k}{l}{r} + \lrlah{n-1}{k-1}{l}{r}.
		\end{equation}
	\end{theorem}

	\begin{proof}
		Let $\pi = (\pi_1, \pi_2, \dots, \pi_l)$ be an $l$-tuple
		of partitions of $[n]$ into $k$ nonempty lists such
		that the numbers $1,2,\dots,r$ are in distinct lists,
		and $\bl \pi_1 = \bl \pi_2 = \dots = \bl \pi_l$. Each
		such $l$-tuple $\pi$ is of exactly one of the following
		two types:

\bigskip
		
\noindent {\it Type A}: There is an $i \in [l]$ such that $n$ is alone in its list in partition $\pi_i$. Then $n$ is a block leader in each of $\pi_1, \pi_2, \dots, \pi_l$; hence $n$ is alone in its list in each of the $\pi_i$. By deleting this list from each of the $\pi_i$, we obtain an $l$-tuple of partitions $\rho = (\rho_1, \rho_2, \dots, \rho_l)$ of $[n-1]$ into $k-1$ nonempty lists such that the numbers $1,2,\dots,r$ are in distinct lists, and $\bl \rho_1 = \bl \rho_2 = \dots = \bl \rho_l$. Going back, by appending the list $(n)$ to each partition $\rho_i$ we recover the initial $l$-tuple $\pi$, hence the number of $l$-tuples $\pi$ of partitions of type A is $\lrlaht{n-1}{k-1}{l}{r}$.

\bigskip

\noindent {\it Type B:}   Otherwise, for each $i \in [l]$, the number $n$ is not alone in its list in $\pi_i$, hence $n$ is not a block leader in any of $\pi_1, \pi_2, \dots, \pi_l$. By deleting $n$ from each of the $\pi_i$, we obtain an $l$-tuple of partitions $\rho = (\rho_1, \rho_2, \dots, \rho_l)$ of the set $[n-1]$ into $k$ nonempty lists such that the numbers $1,2,\dots,r$ are in distinct lists, and $\bl \rho_1 = \bl \rho_2 = \dots = \bl \rho_l$. Going back, by inserting $n$ into some position of some list in each of the $\rho_i$, we obtain an $l$-tuple of partitions of $[n]$ into $k$ nonempty lists such that the numbers $1,2,\dots,r$ are in distinct lists, and the sets of block leaders of these partitions are the same. As the number of possible insertion points in a list equals its length plus one, there are $(n-1)+k = n+k-1$ possible insertion points for $n$ in a single partition $\rho_i$, and $(n+k-1)^l$ possibilities for the $l$-tuple of partitions $\rho$. Hence the number of $l$-tuples $\pi$ of partitions of type $B$ is $(n+k-1)^l\lrlaht{n-1}{k}{l}{r}$.

\bigskip
		
		It follows that the number $\lrlaht{n}{k}{l}{r}$ of all such $l$-tuples $\pi$ of partitions of $[n]$ equals the sum of the number of partitions of type A and the number of partitions of type B, which is $\lrlaht{n-1}{k-1}{l}{r} +\ (n+k-1)^l\lrlaht{n-1}{k}{l}{r}$.
	\end{proof}
	
	This recurrence, together with initial conditions $\lrlaht{n}{k}{l}{r} = 0$ for $n &lt; k$ or $k &lt; r$,  $\lrlaht{n}{n}{l}{r} = 1$, and 
	$\lrlaht{r}{k}{l}{r} = \delta_{k,r}$ can be used to compute the numbers $\lrlaht{n}{k}{l}{r}$ for all $n,k,l,r \ge 0$.
	
	\begin{table}[ht]
		\begin{center}
			\begin{tabular}{|c|c|c|c|c|c|c|c|}
				\hline
				\diagbox[width=2em]{$n$}{$k$} &amp; 2&amp; 3 &amp;4 &amp; 5 &amp; 6 &amp; 7 &amp;8 \\
				\hline
				2 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp;  0 &amp; 0 &amp; 0  \\
				\hline
				3 &amp; 16 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0  \\
				\hline
				4 &amp; 400 &amp; 52 &amp; 1 &amp; 0 &amp; 0 &amp;  0 &amp; 0  \\
				\hline
				5 &amp; 14400 &amp; 2948 &amp; 116 &amp; 1 &amp; 0 &amp; 0 &amp; 0  \\
				\hline
				6 &amp; 705600 &amp; 203072 &amp; 12344 &amp; 216 &amp; 1 &amp; 0 &amp; 0 \\
				\hline
				7 &amp; 45158400 &amp; 17154432 &amp; 1437472 &amp; 38480 &amp; 360 &amp; 1 &amp; 0 \\
				\hline
				8 &amp; 3657830400 &amp; 1760601600 &amp; 191088544 &amp; 6978592 &amp; 99320 &amp; 556 &amp; 1 \\
				\hline
			\end{tabular}
		\end{center}
		\caption{$(2,2)$-Lah numbers $\lrlaht{n}{k}{2}{2}$ for $2 \le n, k\le 8$.}
	\end{table}
	
	\begin{table}[ht]
		\begin{center}
			\begin{tabular}{|c|c|c|c|c|c|c|c|}
				\hline
				\diagbox[width=2em]{$n$}{$k$} &amp; 3 &amp;4 &amp; 5 &amp; 6 &amp; 7 &amp; 8 &amp; 9 \\
				\hline
				3 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp;  0 &amp; 0 &amp; 0  \\
				\hline
				4 &amp; 36 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0  \\
				\hline
				5 &amp; 1764 &amp; 100 &amp; 1 &amp; 0 &amp; 0 &amp;  0 &amp; 0  \\
				\hline
				6 &amp; 112896 &amp; 9864 &amp; 200 &amp; 1 &amp; 0 &amp; 0 &amp; 0  \\
				\hline
				7 &amp; 9144576 &amp; 1099296 &amp; 34064 &amp; 344 &amp; 1 &amp; 0 &amp; 0 \\
				\hline
				8 &amp; 914457600 &amp; 142159392 &amp; 6004512 &amp; 92200 &amp; 540 &amp; 1 &amp; 0 \\
				\hline
				9 &amp; 110649369600 &amp; 21385410048 &amp; 1156921920 &amp; 24075712 &amp; 213700 &amp; 796 &amp; 1 \\
				\hline
			\end{tabular}
		\end{center}
		\caption{$(2,3)$-Lah numbers $\lrlaht{n}{k}{2}{3}$ for $3 \le n, k\le 9$.}
	\end{table}
	
	\begin{proposition}[some special cases]
		\begin{align}
			\lrlah{n}{r}{l}{r}\ &amp;=\ \left(\rlah{n}{r}{r}\right)^{l}\ =\ \frac{(n+r-1)!^{\,l}}{(2r-1)!^{\,l}}, \label{nrrl}\\
			\lrlah{n}{n-1}{l}{r} &amp;=\ 2^l \sum_{j=r}^{n-1} j^l. \label{nn1rl}
		\end{align}
	\end{proposition}
	
	\begin{proof}
		By Definition \ref{lrLah_def}, $\lrlaht{n}{r}{l}{r}$ is the number of ordered $l$-tuples $(\pi_1,\pi_2,\ldots,\pi_l)$ of partitions of $[n]$ into $r$ lists such that the numbers $1,2,\ldots,r$ are in distinct lists, and $\bl \pi_{1} = \bl \pi_{2} = \dots = \bl \pi_{l}$. As there are $r$ lists in each partition, and $1,2,\ldots,r$ are in distinct lists, we have $\bl \pi = [r]$ for all partitions $\pi$, so $\lrlaht{n}{r}{l}{r}$ is simply the $l$-th power of the number of partitions of $[n]$ into $r$ lists such that the numbers $1,2,\ldots,r$ are in distinct lists. Hence $\lrlaht{n}{r}{l}{r} = \left(\rlaht{n}{r}{r}\right)^l$, proving the first equality in (\ref{nrrl}), while the second equality  in (\ref{nrrl}) follows from (\ref{rLahExpl}).
		
		By (\ref{rec1}) with $k = n-1$, we have for $n \ge 1$
		\begin{align}
			\lrlah{n}{n-1}{l}{r}\ &amp;=\ (2n-2)^l \lrlah{n-1}{n-1}{l}{r} +\ \lrlah{n-1}{n-2}{l}{r}\nonumber \\
			&amp;=\ \lrlah{n-1}{n-2}{l}{r} +\ (2n-2)^l \label{2n_2}
		\end{align}
		since $\lrlaht{n-1}{n-1}{l}{r} = 1$. Using (\ref{2n_2}) repeatedly, we obtain
		\begin{align*}
			\lrlah{n}{n-1}{l}{r}\ &amp;=\ \underline{\lrlah{n-1}{n-2}{l}{r}} +\ (2n-2)^l \\
			\underline{\lrlah{n-1}{n-2}{l}{r}}\ &amp;=\ \underline{\lrlah{n-2}{n-3}{l}{r}} +\ (2n-4)^l \\
			&amp;\ \,\vdots \\
			\underline{\lrlah{r+1}{r}{l}{r}}\ &amp;=\ \lrlah{r}{r-1}{l}{r} +\ (2r)^l.
		\end{align*}
		Summing these equations and cancelling all the underlined terms, we find
		\begin{align*}
			\lrlah{n}{n-1}{l}{r}\ &amp;=\ \lrlah{r}{r-1}{l}{r} +\ \sum_{j=r}^{n-1}(2j)^l\ = 2^l \sum_{j=r}^{n-1} j^l,
		\end{align*}
		which is (\ref{nn1rl}).
		
	\end{proof}
	
	\begin{theorem}
		$(l,r)$-Lah numbers can be expressed explicitly as
		\begin{align}
			\lrlah{n}{k}{l}{r}\ &amp;=\!\!\!\!\! \sum_{r+1 \le j_1 &lt; j_2 &lt; \dots &lt; j_{n-k} \le n}
			(2j_1-2)^l (2j_2-3)^l \cdots (2j_{n-k}-(n-k+1))^l.
		\end{align}
	\end{theorem}
	
	\begin{proof} 
		We will compute $\lrlaht{n}{k}{l}{r}$ by counting in how many ways one can construct an $l$-tuple $\pi = (\pi_1,\pi_2,\dots,\pi_l)$ of partitions of the set $[n]$ into $k$ lists such that $1,2,\dots,r$ are in distinct lists, and $\bl \pi_1 = \bl \pi_2 = \dots = \bl \pi_l$. In $\pi_1$ there will be $k$ leading elements $1,2,\dots,r,\lambda_{r+1},\lambda_{r+2}, \dots, \lambda_k \in [n]$ with $r+1 \le \lambda_{r+1} &lt; \lambda_{r+2} &lt; \dots &lt; \lambda_k \le n$, and $n-k$ non-leading elements $\nu_1, \nu_2, \dots, \nu_{n-k} \in [n]$ with $r+1 \le \nu_1 &lt; \nu_2 &lt; \dots &lt;\nu_{n-k} \le n$. 
		
		Starting with a set of $k$ lists of length 1 containing their leaders, i.e., with
		\[
		\{(1), (2), \dots, (r), (\lambda_{r+1}),  (\lambda_{r+2}), \dots, (\lambda_k)\},
		\]
		we insert the non-leading elements $\nu_1, \nu_2, \dots, \nu_{n-k}$ one after another into the above lists, counting the ways to do this as we go along. A non-leading element $\nu_i$ can be inserted into a list iff this list's leader is smaller than $\nu_i$; on the other hand, $\nu_i$ can be inserted into any position within any such list. We now prove that for $i=1,2,\dots,n-k$, \textit{there are exactly $2\nu_i - (i + 1)$ possible insertion points for $\nu_i$}. Recall that when inserting an element into any of $k$ given lists in any position, where the sum of the list lengths is $\ell$, the total number of possible insertion points is $k+\ell$.

		\bigskip
		
\noindent $i = 1$: Since $\nu_1$ is the least non-leader, all the smaller elements $1,2,\dots,\nu_1-1$ are leaders, hence there $\nu_1-1$ lists of length 1 into which $\nu_1$ can be inserted. Hence there are $(\nu_1 - 1) + (\nu_1 - 1) = 2\nu_1 - 2$ possible insertion points for $\nu_1$.

\bigskip
		
\noindent $i &gt; 1$: Assume that all the $\nu_i - 1$ elements smaller than $\nu_i$ have already been inserted into the lists. All of them, except for $\nu_1, \nu_2, \dots, \nu_{i-1}$, are leaders, hence there are $\nu_i - 1 - (i - 1) = \nu_i - i$ lists into which $\nu_i$ can be inserted. These lists jointly contain $\nu_i - 1$ elements, so there are $(\nu_i - i) + (\nu_i - 1) = 2\nu_i - (i + 1)$ possible insertion points for $\nu_i$ as claimed.

\bigskip
		
		In all, we thus have $(2\nu_1 - 2)(2\nu_2 - 3) \cdots (2\nu_{n-k} - (n-k+1))$ ways to construct partition $\pi_1$, and the same number of ways also for each of $\pi_2, \pi_3, \dots, \pi_l$ independently. Hence the number of ways to construct an $l$-tuple $\pi = (\pi_1, \pi_2, \dots, \pi_l)$ for a fixed choice of $\nu_1, \nu_2, \dots, \nu_{n-k}$ is $(2\nu_1 - 2)^l(2\nu_2 - 3)^l \cdots (2\nu_{n-k} - (n-k+1))^l$, and the total number of such $l$-tuples is obtained by summing this product over all possible choices of $\nu_1, \nu_2, \dots, \nu_{n-k}$. Hence
		\begin{align*}
			\lrlah{n}{k}{l}{r}\ &amp;=\!\!\!\!\! \sum_{r+1 \le \nu_1 &lt; \nu_2 &lt; \dots &lt;\nu_{n-k} \le n}
			(2\nu_1-2)^l (2\nu_2-3)^l \cdots (2\nu_{n-k}-(n-k+1))^l,
		\end{align*}
		which proves the theorem.
	\end{proof}
	
	\begin{theorem}
		\label{thmGenF}
		For $k \ge r \ge 1$ and $l \ge 1$, the column generating function
		\[
		F_k^{l,r}(z)\ =\ \sum_{n=0}^\infty \lrlah{n}{k}{l}{r} z^n
		\]
		of $(l,r)$-Lah numbers, as a formal power series, satisfies the difference-differential equation
		\begin{align}
			\label{ddeqF}
			F_k^{l,r}(z)\ &amp;=\ z F_{k-1}^{l,r}(z) + \sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} z^i \frac{d^i}{dz^i}\left(z F_k^{l,r}(z)\right).
		\end{align}
	\end{theorem}
	
	\begin{proof}
		Multiplying recurrence (\ref{rec1}) by $z^n$ and formally summing it on $n$ from $2$ to $\infty$ yields 
		\begin{align}
			\label{genf}
			\sum_{n=2}^\infty \lrlah{n}{k}{l}{r} z^n\ &amp;=\ \sum_{n=2}^\infty (n+k-1)^l \lrlah{n-1}{k}{l}{r} z^n + \sum_{n=2}^\infty \lrlah{n-1}{k-1}{l}{r} z^n.
		\end{align}
		Rewrite the left-hand side of (\ref{genf}) as
		\begin{align*}
			\sum_{n=2}^\infty \lrlah{n}{k}{l}{r} z^n\ =\ F_k^{l,r}(z) - \lrlah{0}{k}{l}{r} - \lrlah{1}{k}{l}{r} z\ =\ \begin{cases}
				F_k^{l,r}(z), &amp; \text{if } k \ge 2; \\
				F_k^{l,r}(z) - z, &amp; \text{if } k = 1.
			\end{cases}
		\end{align*}
		Rewrite the second term on the right-hand side of (\ref{genf}) as
		\begin{align*}
			\sum_{n=2}^\infty \lrlah{n-1}{k-1}{l}{r} z^n\ &amp;=\ \sum_{n=1}^\infty \lrlah{n}{k-1}{l}{r} z^{n+1}\ =\ z \left(F_{k-1}^{l,r}(z) - \lrlah{0}{k-1}{l}{r}\right)\\ &amp;=\ z \begin{cases}
				F_{k-1}^{l,r}(z), &amp; \text{if } k \ge 2;\\
				F_{k-1}^{l,r}(z) - 1, &amp; \text{if } k = 1,
			\end{cases}
			\ =\ \begin{cases}
				z F_{k-1}^{l,r}(z), &amp; \text{if } k \ge 2;\\
				z F_{k-1}^{l,r}(z) - z, &amp; \text{if } k = 1,
			\end{cases}
		\end{align*}
and the first term on the right-hand side of (\ref{genf}) as
		\begin{align*}
\sum_{n=2}^\infty (n+k-1)^l \lrlah{n-1}{k}{l}{r} z^n\ &amp;=\ 
			\sum_{n=2}^\infty \sum_{j=0}^l \binom{l}{j} n^j (k-1)^{l-j} \lrlah{n-1}{k}{l}{r} z^n\\ &amp;=\ 
			\sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{n=2}^\infty \lrlah{n-1}{k}{l}{r} n^j z^n\\ &amp;=\ 
			\sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{n=2}^\infty \lrlah{n-1}{k}{l}{r} \sum_{i=0}^j \sttwo{j}{i} n^{\underline i}\,z^n\\ &amp;=\ 
			\sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} \sum_{n=1}^\infty \lrlah{n}{k}{l}{r} (n+1)^{\underline i}\,z^{n+1}\\ &amp;=\ 
			\sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} \sum_{n=1}^\infty \lrlah{n}{k}{l}{r} z^i \frac{d^i}{dz^i} z^{n+1}\\ &amp;=\ 
			\sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} z^i \frac{d^i}{dz^i}\!\left(z \sum_{n=1}^\infty \lrlah{n}{k}{l}{r} z^n\right)\\ &amp;=\ 
			\sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} z^i \frac{d^i}{dz^i}\!\left(z F_k^{l,r}(z)\right).
		\end{align*}
		Putting the three rewritten terms of equation (\ref{genf}) together again, we obtain
		\begin{align*}
			\begin{cases}
				F_k^{l,r}(z), &amp; \text{if } k \ge 2; \\
				F_k^{l,r}(z) - z, &amp; \text{if } k = 1,
			\end{cases}
			\ &amp;=\ 
			\begin{cases}
				z F_{k-1}^{l,r}(z), &amp; \text{if } k \ge 2;\\
				z F_{k-1}^{l,r}(z) - z, &amp; \text{if } k = 1,
			\end{cases}\\
			&amp;+\ \ \sum_{j=0}^l \binom{l}{j} (k-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} z^i \frac{d^i}{dz^i}\!\left(z F_k^{l,r}(z)\right),
		\end{align*}
		which is equivalent to equation (\ref{ddeqF}).
	\end{proof}
	
	\begin{theorem}
		\label{thmGenP}
		For $n-1 \ge r \ge 1$ and $l \ge 1$, the row generating function
		\begin{align}
			\label{Pnlrx}
			P_n^{l,r}(x) &amp;= \sum_{k=0}^n \lrlah{n}{k}{l}{r} x^k
		\end{align}
		of $(l,r)$-Lah numbers, which is a polynomial in $x$ of degree $n$, satisfies the difference-differential equation
		\begin{align}
			\label{ddeqP}
			P_n^{l,r}(x)\ &amp;=\ x\, P_{n-1}^{l,r}(x) + \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} x^i \frac{d^i}{dx^i}\left(P_{n-1}^{l,r}(x)\right).
		\end{align}
	\end{theorem}
	
	\begin{proof}
		Multiplying recurrence (\ref{rec1}) by $x^k$ and summing it on $k$ from $2$ to $n$ yields 
		\begin{align}
			\label{genp}
			\sum_{k=2}^n \lrlah{n}{k}{l}{r} x^k\ &amp;=\ \sum_{k=2}^n (n+k-1)^l \lrlah{n-1}{k}{l}{r} x^k + \sum_{k=2}^n \lrlah{n-1}{k-1}{l}{r} x^k.
		\end{align}
		Rewrite the left-hand side of (\ref{genp}) as
		\begin{align*}
			\sum_{k=2}^n \lrlah{n}{k}{l}{r} x^k\ &amp;=\ P_n^{l,r}(x) - \lrlah{n}{0}{l}{r} - \lrlah{n}{1}{l}{r} \!\!x.\ % =\ \begin{cases}
				%P_n^{l,r}(x), &amp; n \ge 2, \\
				%P_n^{l,r}(x) - x, &amp; n = 1.
				%\end{cases}
		\end{align*}
		Rewrite the second term on the right-hand side of (\ref{genp}) as
		\begin{align*}
			\sum_{k=2}^n  \lrlah{n-1}{k-1}{l}{r} x^k\ =\ \sum_{k=1}^{n-1} \lrlah{n-1}{k}{l}{r} x^{k+1}\ =\ x \left(P_{n-1}^{l,r}(x) - \lrlah{n-1}{0}{l}{r}\right)
		\end{align*}
and the first term on the right-hand side of (\ref{genp}) as
\begin{align*}
&amp; \quad\quad \sum_{k=2}^n (n+k-1)^l \lrlah{n-1}{k}{l}{r} x^k 
= \sum_{k=2}^n \sum_{j=0}^l \binom{l}{j} k^j (n-1)^{l-j} \lrlah{n-1}{k}{l}{r}x^k
\end{align*}
\begin{align*}
\hphantom{XXXXXXXX} &amp; = \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{k=2}^n \lrlah{n-1}{k}{l}{r} k^j x^k\\ 
\hphantom{XXXXXXXX} &amp; = \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{k=2}^n \lrlah{n-1}{k}{l}{r} \sum_{i=0}^j \sttwo{j}{i} k^{\underline i}\,x^k\\ 
\hphantom{XXXXXXXX} &amp;= \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} \sum_{k=2}^n \lrlah{n-1}{k}{l}{r} k^{\underline i}\,x^k\\ 
\hphantom{XXXXXXXX} &amp;= \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} \sum_{k=2}^n \lrlah{n-1}{k}{l}{r} x^i \frac{d^i}{dx^i} x^k\\ 
\hphantom{XXXXXXXX} &amp; = \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} x^i \frac{d^i}{dx^i}\!\left( \sum_{k=2}^n \lrlah{n-1}{k}{l}{r} x^k \right)\\ 
\hphantom{XXXXXXXX} &amp; = \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} x^i \frac{d^i}{dx^i}\!\left( P_{n-1}^{l,r}(x) - \lrlah{n-1}{0}{l}{r} - \lrlah{n-1}{1}{l}{r}x \right).
\end{align*}
		Putting the three rewritten terms of equation (\ref{genp}) together again, and checking that
		\begin{align*}
			\lrlah{n}{0}{l}{r} + \lrlah{n}{1}{l}{r} &amp;=\ \lrlah{n-1}{0}{l}{r} x\\
			&amp;+\ \sum_{j=0}^l \binom{l}{j} (n-1)^{l-j} \sum_{i=0}^j \sttwo{j}{i} x^i \frac{d^i}{dx^i}\!\left( \lrlah{n-1}{0}{l}{r} + \lrlah{n-1}{1}{l}{r}x \right)
		\end{align*}
		for all $n-1 \ge r \ge 1$ and $l \ge 1$, we obtain equation (\ref{ddeqP}).
	\end{proof}
	
	
	\section{Conclusion}
	\label{five}
	
	
	We conclude the paper with two conjectures, based on experimental evidence.
	
	\begin{conjecture}
		For $n \ge k \ge r \ge 1$, the sequence $\left(\lrlah{n}{k}{l}{r}\right)_{k=0}^n$ is strictly log-concave.
	\end{conjecture}
	
	\begin{conjecture}
		All the roots of polynomials $P_n^{l,r}(x)$, defined in (\ref{Pnlrx}), are real and non-positive.
	\end{conjecture}
	
	\bigskip
	\section{Acknowledgments} 
	
	The author thanks his parents for supporting him, Marko Petkov\v sek for his advice, support and review of the article, and an anonymous referee for advice that improved this article.
	
	
	
	\begin{thebibliography}{99}
		
		
		\bibitem{belbachir2013cross}    H.~Belbachir and A.~Belkhir, Cross recurrence relations for $r$-Lah numbers, \textit{Ars Combin.} \textbf{110} (2013), 199--203.
		
		\bibitem{belbachir2014combinatorial}    H.~Belbachir and I.~E.~Bousbaa, Combinatorial identities for the $r$-Lah numbers, \textit{Ars Combin.} \textbf{115} (2014), 452--458. 
		
		\bibitem{belbachir2021r}    H.~Belbachir and Y.~Djemmada, The $(l,r)$-Stirling numbers: a combinatorial approach, arxiv preprint, arXiv:2101.11039 [math.NT], 2021. Available at \href{http://arxiv.org/abs/2101.11039}{\tt http://arxiv.org/abs/2101.11039}.
		
		\bibitem{broder1984r}    A.~Z.~Broder, The $r$-Stirling numbers, \textit{Discrete Math.} \textbf{49} (1984), 241--259.
		
		\bibitem{carlitz1980_1} 	L.~Carlitz, Weighted Stirling numbers of the first and second kind. I, \textit{Fibonacci Quart.} \textbf{18} (1980), 147--162. 
		
		\bibitem{carlitz1980_2}  	L.~Carlitz, Weighted Stirling numbers of the first and second kind. II, \textit{Fibonacci Quart.} \textbf{18} (1980), 242--257.
		
		\bibitem{cheonjung2012}   G.-S.~Cheon and J.-H.~Jung, $r$-Whitney numbers of Dowling lattices, \textit{Discrete Math.} \textbf{312} (2012), 2337--2348.
		
		\bibitem{GKP89}   R.~L.~Graham, D.~E.~Knuth, and O.~Patashnik, \textit{Concrete Mathematics. A Foundation for Computer Science}, Addison-Wesley, 1989.
		
		\bibitem{kim2022}  D.~S.~Kim, T.~Kim, Central factorial numbers associated with sequences of polynomials, arxiv preprint, arXiv:2202.13130 [math.NT], 2022. Available at  \href{https://arxiv.org/abs/2202.13130}{\tt https://arxiv.org/abs/2202.13130}.
		
		\bibitem{lah1954}   I.~Lah, A new kind of numbers and its application in the actuarial mathematics, \textit{Boletim do Instituto dos Atu\'arios Portugueses} \textbf{9} (1954), 7--15. 
		
		
		\bibitem{nyul2015r}   G.~Nyul and G.~R\'acz, The $r$-Lah numbers, \textit{Discrete Math.} \textbf{338} (2015), 1660--1666.
		
		\bibitem{nyul2021r}   G.~Nyul and G.~R\'acz, Matchings in complete bipartite graphs and the $r$-Lah numbers, \textit{Czechoslovak Math.~J.} \textbf{71} (2021), 947--959.
		
		
		\bibitem{petkovvsek2007combinatorial}   M.~Petkov\v sek and T.~Pisanski, Combinatorial interpretation of unsigned Stirling and Lah numbers, \textit{Pi Mu Epsilon J.} \textbf{12} (2007), 417--424.
		
		\bibitem{riordan1958}  J.~Riordan, \textit{An Introduction to Combinatorial Analysis},  John Wiley \&amp; Sons, Inc.; Chapman \&amp; Hall, Ltd., 1958.
		
		\bibitem{shattuck2014arXiv}  M.~Shattuck, Generalized $r$-Lah numbers, arxiv preprint, arXiv:1412.8721 [math.NT], 2014. Available at \href{http://arxiv.org/abs/1412.8721}{\tt http://arxiv.org/abs/1412.8721}.
		
		\bibitem{shattuck2016}  M.~Shattuck, Generalized $r$-Lah numbers, \textit{Proc.\ Indian Acad.\ Sci.\ Math.\ Sci.} \textbf{126} (2016), 461--478. 
	\end{thebibliography} 
	
	\bigskip
	\hrule
	\bigskip
	
	\noindent 
	2020 \emph{Mathematics Subject Classification}:~Primary 05A19. 
	Secondary 05A18, 11B37, 11B73.
	
	\medskip
	
	\noindent 
	\emph{Keywords}:~ Lah numbers, 
	Stirling number of the first and second kind, 
	$r$-Lah number, 
	$r$-Stirling number of the first and second kind, 
	$(l,r)$-Lah number, 
	$(l,r)$-Stirling number of the first and second kind, 
	recurrence relation, 
	explicit representation,
	generating function,
	$(l,r)$-Lah polynomial.
	
	\bigskip
	\hrule
	\bigskip
	
\noindent 
(Concerned with sequences  
\seqnum{A008277},
\seqnum{A008297},
\seqnum{A132393},
\seqnum{A143497},
\seqnum{A143498}, and
\seqnum{A143499}.)
	
	\bigskip
	\hrule
	\bigskip
	
	\vspace*{+.1in}
	\noindent
	Received June 27 2022;
	revised version received January 25 2023.
	Published in {\it Journal of Integer Sequences},  February 25 2023.
	
	\bigskip
	\hrule
	\bigskip
	
	\noindent
	Return to
	\href{https://cs.uwaterloo.ca/journals/JIS/}{Journal of Integer Sequences home page}.
	\vskip .1in
	
\end{document}


</pre></body></html>