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\begin{center}
\vskip 1cm{\LARGE\bf
Combinatorial Properties of a Generalized Class of Laguerre Polynomials\\
\vskip .1in } \vskip 1cm \large
Mark Shattuck\\
Department of Mathematics\\
University of Tennessee\\
Knoxville, TN 37996\\
USA\\
\href{mailto:shattuck@math.utk.edu}{\tt shattuck@math.utk.edu}\\
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\begin{abstract}
In this paper, we consider various combinatorial aspects of a
family of polynomials, denoted by $L_n^{(\alpha,\beta)}(x)$, whose
coefficients $S_{\alpha,\beta}(n,k)$ correspond to a special case
of the partial $r$-Bell polynomials. Among the particular cases of
$L_n^{(\alpha,\beta)}(x)$ are the generalized Laguerre polynomials,
associated Lah polynomials, and polynomials arising in the study
of hyperbolic partial differential equations. Here we provide a
combinatorial treatment of $L_n^{(\alpha,\beta)}(x)$ and its coefficients,
which were studied previously strictly from an algebraic standpoint.
In addition to providing combinatorial proofs of some prior identities, we
derive several new relations using the combinatorial interpretations for
$L_n^{(\alpha,\beta)}(x)$ and $S_{\alpha,\beta}(n,k)$. Our proofs make
frequent use of sign-changing involutions on various weighted structures.
Finally, we introduce a bivariate polynomial generalization arising
as a distribution for a pair of statistics and establish some of its
basic properties.
\end{abstract}
\section{Introduction}
The generalized sequence of polynomials $L_n^{(\alpha,\beta)}(x)$ are defined by Mihoubi and Sahari \cite{MS} as coefficients in the exponential generating function formula
\begin{equation}\label{introe1}
\sum_{n\geq0}L_n^{(\alpha,\beta)}(x)\frac{t^n}{n!}=(1-t)^\alpha \exp\left(x\left((1-t)^\beta-1\right)\right),
\end{equation}
where $\alpha$ and $\beta$ are real numbers with $\beta \neq 0$. The special cases of $L_n^{(\alpha,\beta)}(x)$ when $(\alpha,\beta)=(-\frac{1}{2},-\frac{1}{2})$ or $(-\frac{3}{2},-\frac{1}{2})$ arise in the theory of hyperbolic partial differential equations, see, \cite[pp.\ 391--398]{CH} and \cite{KD}. Other important special cases include the generalized Laguerre polynomials ($\alpha=-\lambda-1, \beta=-1$) and the associated Lah polynomials ($\alpha=0,\beta=-m$), see, e.g., \cite{D} and \cite{AE}, respectively. It is seen from \eqref{introe1} that $L_n^{(\alpha,\beta)}(x)$ may be expressed as
$$L_n^{(\alpha,\beta)}(x)=\sum_{k=0}^nS_{\alpha,\beta}(n,k)x^k,$$
where
\begin{equation}\label{introe2}
S_{\alpha,\beta}(n,k)=\frac{1}{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}(-\alpha-\beta j)^{\overline{n}}, \qquad n,k \geq0,
\end{equation}
and $x^{\overline{n}}=x(x+1)\cdots(x+n-1)$ if $n\geq 1$, with $x^{\overline{0}}=1$, denotes the rising factorial. Note that \eqref{introe2} implies $S_{\alpha,\beta}(n,k)=0$ for $k>n\geq 0$, whence the $L_n^{(\alpha,\beta)}(x)$ are indeed polynomials.
Recall that the partial $r$-Bell polynomials \cite{MR} are defined by
$$\sum_{n\geq k}B_{n+r,k+r}^{(r)}(a_\ell;b_\ell)\frac{t^n}{n!}=\frac{1}{k!}\left(\sum_{j\geq 1}a_j\frac{t^j}{j!}\right)^k\left(\sum_{j\geq0}b_{j+1}\frac{t^j}{j!}\right)^r,$$
the $r=0$ case of which corresponds to the classical partial Bell polynomials (see, e.g., \cite{Bell}).
It was shown \cite{MS} that
\begin{equation}\label{introe3}
S_{r\alpha,\beta}(n,k)=B_{n+r,k+r}^{(r)}\left((-\beta)^{\overline{j}},(-\alpha)^{\overline{j-1}}\right),
\end{equation}
where $r$ is a non-negative integer. Note that upon taking $\alpha=\beta=-1$ in \eqref{introe3} and replacing $r$ with $2r$, one obtains the $r$-Lah numbers as a special case of $S_{\alpha,\beta}(n,k)$, which we denote by $\Lah{n}{k}_r$ in accordance with \cite{NR}. The $r=0$ case of $\Lah{n}{k}_r$, which is written without a subscript, corresponds to the classical Lah numbers; see, e.g., \cite{Lah} and \seqnum{A008297} in \cite{Slo}. We remark that other polynomial generalizations of the Lah numbers related to $S_{\alpha,\beta}(n,k)$ have been considered previously \cite{MSS,Sh}.
Here, we provide a unified combinatorial treatment of several of the previous algebraic results involving $L_n^{(\alpha,\beta)}(x)$ and $S_{\alpha,\beta}(n,k)$. To do so, we consider a slight variant, denoted by $L_r(n,k)=L_r^{(\alpha,\beta)}(n,k)$, of $S_{\alpha,\beta}(n,k)$ defined as $L_r(n,k)=S_{-r\alpha,-\beta}(n,k)$. We then let $L_{n,r}(x)=L_{n,r}^{(\alpha,\beta)}(x)$ be given by
\begin{equation}\label{defeqn}
L_{n,r}(x)=\sum_{k=0}^n L_r(n,k)x^k, \qquad n \geq 0.
\end{equation}
We find these forms of $S_{\alpha,\beta}(n,k)$ and $L_n^{(\alpha,\beta)}(x)$ to be more convenient to deal with combinatorially. The extra parameter $r$ is introduced here since in several of the subsequent identities, we consider integral increments of the parameter $\alpha$. Also, in the final section, the non-negative integer $r$ is replaced by its polynomial analogue $\frac{p^r-1}{p-1}$, where $p$ is an indeterminate.
The organization of this paper is as follows. In the next section, we find a combinatorial interpretation for $L_r(n,k)$ and $L_{n,r}(x)$ in terms of a pair of statistics on a structure closely related to the $r$-Lah distributions. We make use of this interpretation in the third section in finding some new relations involving $L_r(n,k)$ and $L_{n,r}(x)$. In the fourth section, we provide combinatorial proofs of some prior formulas for $S_{\alpha,\beta}(n,k)$ and $L_n^{(\alpha,\beta)}(x)$, rewritten in terms of $L_r(n,k)$ and $L_{n,r}(x)$, which were found previously by algebraic methods. Our proofs entail use of weight-preserving, sign-reversing involutions defined on certain weighted configurations involving various kinds of finite partitions whose blocks are contents-ordered. In the final section, we consider a ($p,q$)-generalization of $L_r(n,k)$ (and hence also of $L_{n,r}(x)$) by considering two further statistics (marked by $p$ and $q$) on the underlying structure that is enumerated by $L_r(n,k)$ when $\alpha=\beta=1$. Some identities are found of the ($p,q$)-analogue, which extend earlier ones and the log-concavity is established for a range of $p$ and $q$ values.
\section{Combinatorial definition and generating function}
In this section, we provide a combinatorial interpretation for the sequences $L_r(n,k)$ and $L_{n,r}(x)$ and show how their (exponential) generating function formulas can be obtained using this definition. To do so, we first write a two-term recurrence for $L_r(n,k)$. Such a recurrence for $L_r(n,k)$ where $n,k \geq 1$ (along with initial conditions) may be derived from \eqref{introe2} using $L_r(n,k)=S_{-r\alpha,-\beta}(n,k)$ as follows:
\begin{align*}
&\beta L_r(n-1,k-1)+(\alpha r+\beta k+n-1)L_r(n-1,k)\\
&\qquad\qquad=\frac{\beta}{(k-1)!}\sum_{j=0}^{k-1}(-1)^{k-1-j}\binom{k-1}{j}(\alpha r+\beta j)^{\overline{n-1}}\\
&\qquad\qquad\quad+\frac{\alpha r+\beta k+n-1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}(\alpha r+\beta j)^{\overline{n-1}}\\
&\qquad\qquad=\frac{\beta}{k!}\sum_{j=0}^{k-1}(-1)^{k-1-j}(k-j)\binom{k}{j}(\alpha r+\beta j)^{\overline{n-1}} +\frac{(\alpha r+\beta k)^{\overline{n}}}{k!}\\
&\qquad\qquad\quad+\frac{\alpha r+\beta k+n-1}{k!}\sum_{j=0}^{k-1}(-1)^{k-j}\binom{k}{j}(\alpha r+\beta j)^{\overline{n-1}}\\
&\qquad\qquad=\frac{1}{k!}\sum_{j=0}^{k-1}(-1)^{k-j}\binom{k}{j}(\alpha r+\beta j)^{\overline{n-1}}\left(\beta(j-k)+\alpha r +\beta k+n-1\right)
+\frac{(\alpha r+\beta k)^{\overline{n}}}{k!}
\end{align*}
\begin{align*}
&=\frac{1}{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}(\alpha r+\beta j)^{\overline{n}}=L_r(n,k).
\end{align*}
Thus, we have the recurrence
\begin{equation}\label{defrec}
L_r(n,k)=\beta L_r(n-1,k-1)+(\alpha r+\beta k+n-1)L_r(n-1,k), \qquad n,k \geq 1,
\end{equation}
with initial conditions $L_r(n,0)=(\alpha r)^{\overline{n}}$ and $L_r(0,k)=\delta_{k,0}$ for all $n,k \geq 0$.
Note that when $\alpha=\beta=1$ and $r$ is replaced by $2r$, it is apparent from \eqref{defrec} that $L_r(n,k)$ reduces to the $r$-Lah number. For a different generalization of the $r$-Lah numbers, see \cite{Cheon}, where the $r$-Whitney-Lah numbers are introduced, and \cite{GNy,RamSh}, where they are studied further.
We now find a combinatorial interpretation for $L_r(n,k)$ in terms of a bivariate distribution. Given $n,k,r \geq 0$, let $\mathcal{P}_{n,k}=\mathcal{P}_{n,k}^{(r)}$ denote the set of partitions of $[n+r]$ into $k+r$ contents-ordered blocks such that the elements of $[r]$ belong to distinct blocks, with these elements first in their respective blocks. Within $\lambda \in \mathcal{P}_{n,k}$, we refer to blocks containing the elements of $[r]$ as \emph{special}, with all other blocks being \emph{non-special}, and at times refer to the elements of $[r]$ themselves as special. Note that by virtue of their association with a distinguished element, special blocks are labeled and in effect allowed to be empty, whereas non-special blocks are unlabeled and always non-empty.
We define a pair of statistics on $\mathcal{P}_{n,k}$ as follows. By a left-right minimum (lr min) within a non-special block $B=\{b_1b_2\cdots b_\ell\}$ of $\lambda$, we mean an element $b_i \in B$ such that $b_in$ or $k<0$.
We have the following combinatorial interpretation for $L_r(n,k)$.
\begin{theorem}\label{Combth}
For all $n,k,r\geq0$, $L_r(n,k)=T_r(n,k)$; i.e., $L_r(n,k)$ is the joint distribution for the statistics $\nu_1$ and $\nu_2$ on $\mathcal{P}_{n,k}^{(r)}$.
\end{theorem}
\begin{proof}
The initial conditions of $T_r(n,k)$ when $n=0$ or $k=0$ are seen to agree with those of $L_r(n,k)$, so assume $n,k \geq 1$. Note that the weight of the members of $\mathcal{P}_{n,k}$ in which the element $n+r$ comprises its own block is given by $\beta T_r(n-1,k-1)$. On the other hand, if $n+r$ is placed just after a member of $[r]$ within one of the special blocks or at the beginning of a non-special block, then there are $\alpha rT_r(n-1,k)$ and $\beta k T_r(n-1,k)$ possibilities, respectively. Finally, members of $\mathcal{P}_{n,k}$ in which $n+r$ directly follows some element of $[r+1,r+n-1]$ contribute $(n-1)T_r(n-1,k)$ towards the weight. Combining the previous cases implies $T_r(n,k)$ satisfies recurrence \eqref{defrec}, and hence $T_r(n,k)=L_r(n,k)$ for all $n$, $k$ and $r$.
\end{proof}
Let $\mathcal{P}_n=\mathcal{P}_n^{(r)}$ be given by $\mathcal{P}_n=\cup_{k=0}^n\mathcal{P}_{n,k}$ for $n \geq 0$. Given $\lambda \in \mathcal{P}_n$, let $\mu(\lambda)$ denote the number of non-special blocks of $\lambda$. Then, from \eqref{defeqn}, it is seen that $L_{n,r}(x)$ gives the joint distribution of the $\nu_1$, $\nu_2$ and $\mu$ statistics on $\mathcal{P}_n$, where the $\mu$ statistic is marked by the $x$ variable. Note that $L_{n,r}(x)$ reduces when $x=\alpha=\beta=1$ to the $n$-th row sum of $r$-Lah numbers, which coincides with \seqnum{A000262} when $r=0$.
The $L_n^{(\alpha,\beta)}(x)$, equivalently the $L_{n,r}(x)$, were defined in \cite{MS} as coefficients of a certain exponential generating function (egf), from which various algebraic properties are derived. Alternatively, starting with the combinatorial definition above for $L_{n,r}(x)$, it is possible to derive the corresponding egf formula.
\begin{theorem}\label{egf}
We have
\begin{equation}\label{egfe1}
\sum_{n\geq0}L_{n,r}(x)\frac{t^n}{n!}=(1-t)^{-\alpha r}\exp\left(x\left((1-t)^{-\beta}-1\right)\right)
\end{equation}
and
\begin{equation}\label{egfe2}
\sum_{n\geq k}L_r(n,k)\frac{t^n}{n!}=\frac{(1-t)^{-\alpha r}}{k!}\left((1-t)^{-\beta}-1\right)^k.
\end{equation}
\end{theorem}
\begin{proof}
Since \eqref{egfe2} follows from \eqref{egfe1} and \eqref{defeqn}, we need only establish \eqref{egfe1}. To do so, first observe the identity
\begin{equation}\label{P_nbrec}
L_{n+1,r}(x)=(n+\alpha r+\beta x)L_{n,r}(x)+\beta x \frac{\partial}{\partial x}L_{n,r}(x), \qquad n \geq 0.
\end{equation}
An equivalent form of \eqref{P_nbrec} was shown in \cite{MS} algebraically. Using the interpretation given above for $L_{n,r}(x)$, one can give a quick combinatorial proof of \eqref{P_nbrec} as follows. Note that the first term on the right side of \eqref{P_nbrec} counts all $\rho \in \mathcal{P}_{n+1}$
where the element $n+r+1$ either directly follows a member of $I$, follows a special element or occurs as a singleton block. The second term is seen to count those $\rho$ in which $n+r+1$ starts a non-singleton non-special block (and hence itself is an lr min). Observe that since no new block is created in this last case, no factor of $x$ is introduced, which is witnessed with the multiplication of the $x$-partial derivative by $x$.
Let $f(t,x)=\sum_{n\geq 0}L_{n,r}(x)\frac{t^n}{n!}$. Multiplying both sides of \eqref{P_nbrec} by $\frac{t^n}{n!}$, and summing over $n \geq 0$, gives
\begin{equation}\label{pde}
(1-t)\frac{\partial}{\partial t}f(t,x)-\beta x\frac{\partial}{\partial x}f(t,x)=(\alpha r+\beta x)f(t,x),
\end{equation}
with initial condition $f(0,x)=1$. Solving explicitly the first-order linear partial differential equation \eqref{pde} then yields
$$f(t,x)=(1-t)^{-\alpha r}\exp\left(x\left((1-t)^{-\beta}-1\right)\right),$$
as desired.
\end{proof}
\begin{remark}
Note that recurrence \eqref{defrec} also follows from equating like powers of $x$ on both sides of \eqref{P_nbrec}.
\end{remark}
\section{New identities for $L_r(n,k)$ and $L_{n,r}(x)$}
In this section, we derive some new identities involving $L_r(n,k)$ and $L_{n,r}(x)$. Applying recurrence \eqref{defrec} repeatedly yields the following formula for $n \geq k \geq 1$:
\begin{equation}\label{defrece1}
L_r(n,k)=(\alpha r)^{\overline{n-k}}\beta^k+\sum_{j=0}^{k-1}(\alpha r+\beta(k-j)+n-j-1)\beta^jL_r(n-j-1,k-j),
\end{equation}
which may also be shown by considering the largest element $n-j+r$ that either goes in a special block or in a non-singleton non-special block. Multiplying both sides of \eqref{defrece1} by $x^k$, and summing over $1 \leq k \leq n$, gives after simplification the following recurrence:
\begin{equation}\label{defrece2}
L_{n,r}(x)=(\beta x)^n+\sum_{j=0}^{n-1}(\beta x)^{n-j-1}\left((\alpha r+j)L_{j,r}(x)+\beta x \frac{\partial}{\partial x}L_{j,r}(x)\right), \qquad n \geq 1.
\end{equation}
From \eqref{introe2}, $L_r(n,k)$ is given explicitly by
\begin{equation}
L_r(n,k)=\frac{1}{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}(\alpha r+\beta j)^{\overline{n}}, \qquad n,k \geq0. \label{exforme1}
\end{equation}
Let $x^{\underline{n}}=x(x-1)\cdots(x-n+1)$ for $n \geq 1$, with $x^{\underline{0}}=1$, denote the falling factorial. We have the following further recurrences for $L_r(n,k)$ that can be obtained from \eqref{exforme1}.
\begin{theorem}\label{furecs}
If $n,k \geq 0$ and $0 \leq s \leq r$, then
\begin{equation}\label{furecse1}
L_r^{(\alpha,\beta)}(n,k)=\sum_{j=k}^n\binom{n}{j}(\alpha(r-s))^{\overline{n-j}}L_s^{(\alpha,\beta)}(j,k)
\end{equation}
and
\begin{equation}\label{furecse2}
L_r^{(\alpha,\alpha)}(n,k)=\sum_{j=k}^n\binom{j}{k}(r-s)^{\underline{j-k}}L_s^{(\alpha,\alpha)}(n,j).
\end{equation}
\end{theorem}
\begin{proof}
By \eqref{exforme1}, we have
\begin{align*}
&\sum_{j=k}^n\binom{n}{j}(\alpha(r-s))^{\overline{n-j}}L_s^{(\alpha,\beta)}(j,k)\\
&=\sum_{j=0}^n\binom{n}{j}(\alpha(r-s))^{\overline{n-j}}\cdot \frac{1}{k!}\sum_{\ell=0}^k (-1)^{k-\ell}\binom{k}{\ell}(\alpha s +\beta \ell)^{\overline{j}}\\
&=\frac{1}{k!}\sum_{\ell=0}^k(-1)^{k-\ell}\binom{k}{\ell}\sum_{j=0}^n\binom{n}{j}(\alpha(r-s))^{\overline{n-j}}(\alpha s +\beta \ell)^{\overline{j}}\\
&=\frac{1}{k!}\sum_{\ell=0}^k(-1)^{k-\ell}\binom{k}{\ell}(\alpha r +\beta \ell)^{\overline{n}}=L_r^{(\alpha,\beta)}(n,k),
\end{align*}
where we have used the fact $(x+y)^{\overline{n}}=\sum_{j=0}^n \binom{n}{j}x^{\overline{j}}y^{\overline{n-j}}$ in the penultimate equality. For \eqref{furecse2}, first observe that for a fixed $n$, $k$ and $s$ where $n \geq k$, both sides of \eqref{furecse2} may be viewed as polynomials in $r$ of degree $n-k$. Thus, it suffices to show \eqref{furecse2} for all $r \geq n-k+s$. Let $m=k+r-s$. By \eqref{exforme1}, we then have
\begin{align*}
&\sum_{j=k}^n\binom{j}{k}(r-s)^{\underline{j-k}}L_s^{(\alpha,\alpha)}(n,j)=\sum_{j=k}^m\binom{j}{k}(r-s)^{\underline{j-k}}\cdot \frac{1}{j!}\sum_{\ell=0}^j(-1)^{j-\ell}\binom{j}{\ell}(\alpha(s+\ell))^{\overline{n}}\\
&=\sum_{\ell=0}^m(\alpha(s+\ell))^{\overline{n}}\sum_{j=k}^m\frac{(-1)^{j-\ell}}{j!}\binom{j}{k}\binom{j}{\ell}(r-s)^{\underline{j-k}}\\
&=\frac{1}{k!}\sum_{\ell=0}^m(\alpha(s+\ell))^{\overline{n}}\sum_{j=k}^m(-1)^{j-\ell}\binom{r-s}{j-k}\binom{j}{\ell}\\
&=\frac{1}{k!}\sum_{\ell=0}^m(\alpha(s+\ell))^{\overline{n}}\cdot(-1)^{r-s+k-\ell}\binom{k}{\ell-r+s}\\
&=\frac{1}{k!}\sum_{\ell=0}^k(-1)^{k-\ell}\binom{k}{\ell}(\alpha(r+\ell))^{\overline{n}}=L_r^{(\alpha,\alpha)}(n,k),
\end{align*}
where we have used \cite[Formula~5.24]{GKP} in the antepenultimate equality
\end{proof}
\begin{remark} Identities \eqref{furecse1} and \eqref{furecse2}, in the case when $\alpha=\beta=1$ and $r$ and $s$ are even, were shown in \cite{NR} by a different method. Furthermore, identity \eqref{furecse1} may be given a combinatorial proof by considering the number $n-j$ of elements of $I$ in the final $r-s$ special blocks within a member of $\mathcal{P}_{n}^{(r)}$. Identity \eqref{furecse2} may also be obtained combinatorially by first arranging the elements of $[n+s]$ according to a member of $\mathcal{P}_{n,j}^{(s)}$ where $j \geq k$ and then selecting $j-k$ of the non-special blocks whose contents to be transferred to $r-s$ additional special blocks. We leave the details of this argument to the interested reader.
\end{remark}
The combinatorial interpretation for $L_r(n,k)$ given in Theorem \ref{Combth} above yields further recurrence formulas for $L_r(n,k)$ as follows.
\begin{theorem}\label{neqns}
If $n,m,k\geq 0$ and $0 \leq s \leq r$, then
\begin{equation}\label{neqnse1}
L_r(n+m,k)=\sum_{i=0}^n\sum_{j=0}^m\binom{n}{i}\binom{m}{j}(\alpha(r-s))^{\overline{i+j}}L_s(n+m-i-j,k)
\end{equation}
and
\begin{align}
L_r(n+m+1,k)=\sum_{i=0}^n\sum_{j=0}^m&\binom{n}{i}\binom{m}{j}\big(r\alpha^{\overline{i+j+1}}L_{r-1}(n+m-i-j,k)\notag\\
&+\beta^{\overline{i+j+1}}L_r(n+m-i-j,k-1)\big).\label{neqnse2}
\end{align}
\end{theorem}
\begin{proof}
To show \eqref{neqnse1}, consider the number $i$ of elements of $I$ and the number $j$ of elements of $[r+n+1,r+n+m]$ that go in the final $r-s$ special blocks within a member of $\mathcal{P}_{n+m,k}$. There are $\binom{n}{i}\binom{m}{j} (\alpha(r-s))^{\overline{i+j}}$ ways in which to choose and arrange these elements and then $L_s(n+m-i-j,k)$ ways to arrange the remaining members of $[r+1,r+n+m]$, together with the first $s$ special elements. Summing over all possible $i$ and $j$ gives \eqref{neqnse1}.
For \eqref{neqnse2}, we consider the number $i$ of elements of $[r+2,r+n+1]$ and the number $j$ of elements of $[r+n+2,r+n+m+1]$ that go in the same block as $r+1$ within a member of $\mathcal{P}_{n+m+1,k}$. If $r+1$ is to go in one of the $r$ special blocks, then there are $\alpha^{\overline{i+j+1}}$ ways in which to order the elements in this block and $L_{r-1}(n+m-i-j,k)$ ways in which to arrange the remaining members of $[r+n+m+1]$. If $r+1$ goes in a non-special block, then there are $\beta^{\overline{i+j+1}}$ ways to arrange the elements in this block and $L_r(n+m-i-j,k-1)$ ways in which to arrange the remaining elements. Combining the two previous cases gives the generic term in the sum on the right side of \eqref{neqnse2}, which implies the result.
\end{proof}
A similar argument to that given for \eqref{neqnse1} above implies
$$L_{n+m,r}(x)=\sum_{i=0}^n\sum_{j=0}^m\binom{n}{i}\binom{m}{j}(\alpha(r-s))^{\overline{i+j}}L_{n+m-i-j,s}(x), \qquad 0 \leq s \leq r,$$
which may also be obtained by multiplying both sides of \eqref{neqnse1} by $x^k$ and summing over all $k$. A formula comparable to \eqref{neqnse2} may be given for $L_{n+m+1,r}(x)$ as well. The $m=r=0$ case of \eqref{neqnse2} may be written equivalently as
$$L_0(n+1,k)=\sum_{i=0}^n \binom{n}{i}\beta^{\overline{i+1}}L_0(n-i,k-1), \qquad n,k \geq 0,$$
which can also be obtained by considering the number of additional elements in the block containing $1$. When $\beta=1$, note that this is a standard recurrence for $\Lah{n}{k}$.
\begin{theorem}\label{L(n,k)rec}
If $n,m,j\geq 0$ and $0 \leq s \leq r$, then
\small \begin{align}
L_r(n+1,m+j+1)=\beta\sum_{i=0}^{n-m-j}\sum_{k=m}^{n-i}\sum_{\ell=j}^{n-i-k}&\binom{n+1}{i}\binom{n-i-k}{\ell}(\alpha s+\beta(m+1)+k+1)^{\overline{n-i-k-\ell}}\label{L(n,k)rece1}\\
&\times (\alpha(r-s))^{\overline{i}}L_s(k,m)L_0(\ell,j). \notag
\end{align}\normalsize
\end{theorem}
\begin{proof}
We enumerate the members of $\mathcal{P}_{n+1,m+j+1}$ according to $i$, $k$ and $\ell$ defined as follows. Given $\lambda \in \mathcal{P}_{n+1,m+j+1}$, let $i$ denote the number of elements of $[r+1,r+n+1]$ going in the final $r-s$ special blocks of $\lambda$, and let $R$ denote the subset of elements so selected. Let $T=[r+1,r+n+1]-R$ be given by $T=\{t_1,\ldots,t_{n-i+1}\}$ where $t_1<\cdots1$, it is seen that there are
$\alpha s+\beta i+m+i-1$ ways in which to insert $x_i$, since in addition to the possibilities mentioned for $x_1$, the element $x_i$ may be placed directly after any member of $x_1,\ldots,x_{i-1}$.
Thus, there are $(\alpha s +\beta i+m)^{\overline{n-j}}$ ways in which to insert the elements of $X$ into the previous blocks. Further, a member of $T-X$ can go either in a non-special block containing no members of $R$ or in any one of the final $r-s$ special blocks such that no element of $R$ occurs between it and the special element in that block. There are then $L_{j,r-s}(x)$ ways in which to arrange the elements of $T-X$ in their blocks as there is no restriction on the number of additional non-special blocks that are to be occupied. Note that members of $T$ belonging to one of the final $r-s$ special blocks and occurring to the right of the leftmost member of $R$ in the block (if it exists) all belong to $X$, while those occurring to the left belong to $T-X$ (with all members belonging to $T-X$ if it is the case that the block contains no member of $R$). Therefore, given $\lambda \in \mathcal{P}_{m+n}$, the elements of $T-X$, and hence of $X$, may be retrieved by considering the contents of the final $r-s$ special blocks together with any non-special blocks that fail to contain an element of $R$. Considering all possible values of $i$ and $j$ thus implies the result.
\end{proof}
\begin{remark} The case of \eqref{P{n,r}forme1} where $\alpha=\beta=1$ and $r,s$ are both even corresponds to \cite[Theorem~3.3]{NR2}, which was shown algebraically by finding two different expansions of $(x+2r)^{\overline{m+n}}$ and equating like coefficients of $x^{\underline{k}}$.
\end{remark}
Proceeding as in the proof of \cite[Theorem~3.1]{Sh2} yields the following further formula, where $n,k,r \geq 0$ and $x_i$ and $y_i$ for $i \geq 0$ denote arbitrary sequences:
\begin{align}
\sum_{j=0}^{n+k}\binom{n+k}{j}x_jy_{n+k-j}L_{j,r}(x)=\sum_{p=0}^k\sum_{\ell=0}^k\sum_{j=0}^n\sum_{i=0}^j&\binom{n}{j}\binom{k}{\ell}\binom{j}{i}x_{n+\ell+i-j}y_{k+j-\ell-i}\notag\\
&\times x^p(\alpha r+\beta p+\ell)^{\overline{n-j}}L_r(\ell,p)L_{i,0}(x). \label{Spie1}
\end{align}
We note an important special case of \eqref{Spie1}. Let $U_{n,r}(x)=\sum_{m=0}^n\Lah{n}{m}_rx^m$ be the $r$-Lah polynomial of order $n$, the $r=0$ case of which being denoted simply by $U_n(x)$. Taking $x_i=1$ and $y_i=\delta_{i,0}$ for all $i \geq 0$ in \eqref{Spie1}, with $\alpha=\beta=1$, gives the formula
\begin{equation}\label{Spie2}
U_{n+k,r}(x)=\sum_{p=0}^k\sum_{j=0}^n\binom{n}{j}x^p(k+p+2r)^{\overline{n-j}}\Lah{k}{p}_rU_{j}(x),
\end{equation}
which is an $r$-Lah polynomial version of Spivey's formula \cite{Sp} for the classical Bell numbers \seqnum{A000110}.
\section{Combinatorial proofs of prior identities}
In this section, we provide combinatorial proofs of some prior formulas involving $L_r(n,k)$ and $L_{n,r}(x)$ that were shown previously by various algebraic methods.
We first prove a couple of relations involving $L_{n,r}(x)$ and the two kinds of Stirling numbers that occur in a slightly different form as \cite[Proposition~5]{MS}. Let $\stirling{n}{k}$ denote the (signless) Stirling number of the first kind \seqnum{A008275} and $\Stirling{n}{k}$ the Stirling number of the second kind \seqnum{A008277}. Let $B_n(x)=\sum_{k=0}^n \Stirling{n}{k}x^k$ be the $n$-th Bell polynomial (see, e.g., \cite{MStext} or \cite{MiB}).
\begin{theorem}\label{eqnpair}
If $n,r \geq 0$, then
\begin{equation}\label{eqnpaire1}
L_{n,r}(x)=\sum_{k=0}^n\sum_{j=0}^k\stirling{n}{k}\binom{k}{j}(\alpha r)^{k-j}\beta^jB_j(x)
\end{equation}
and
\begin{equation}\label{eqnpaire2}
\sum_{k=0}^n (-1)^{n-k}\Stirling{n}{k}L_{k,r}(x)=\sum_{k=0}^n\binom{n}{k}(\alpha r)^{n-k}\beta^kB_k(x).
\end{equation}
\end{theorem}
\begin{proof}
To show \eqref{eqnpaire1}, we form members $\lambda \in \mathcal{P}_n$ as follows. First arrange the elements of $I$ according to a permutation of $[n]$ having exactly $k$ cycles expressed in standard cycle form (i.e., smallest element first within each cycle, with cycles arranged from left to right in increasing order of first elements). Let $C_1,C_2,\ldots,C_k$ denote the cycles so obtained. We then select $k-j$ of the $C_i$, and for each chosen $C_i$, we write the elements contained therein in the same order within one of the special blocks (after the special element). If two or more $C_i$ are selected for the same special block, say $C_{i_1},C_{i_2},\ldots,$ where $i_1>i_2>\cdots$, then we write all of the element contained in $C_{i_1}$ first, followed by those in $C_{i_2}$ and so on. Then there are $\binom{k}{j}(\alpha r)^{k-j}$ ways in which to select the $k-j$ cycles and arrange the elements contained therein. Note that the first element of each of the chosen $C_i$ corresponds to a special block lr min and hence contributes a factor of $\alpha$ towards the weight. The remaining $j$ cycles are then arranged according to an arbitrary partition of $[j]$ such that the contents of cycles going in the same block are written in descending order of their indices. Thus each cycle starter in this case corresponds to a non-special block lr min, which accounts for the factor of $\beta^j$. This aside, there are then $B_j(x)$ possibilities concerning the relative positions of the cycles within the various (non-special) blocks. Considering all $k$ and $j$ then yields uniquely all possible $\lambda$, which implies \eqref{eqnpaire1}.
To show \eqref{eqnpaire2}, first let $\mathcal{A}_{n,k}$ for $1 \leq k \leq n$ denote the set of ordered pairs $(\pi,\rho)$, where $\pi$ is a partition of $I$ with $k$ blocks and $\rho$ is a member of $\mathcal{P}_k$ using the blocks of $\pi$ as its non-special elements (together with the members of $[r]$), where blocks are ordered by the relative sizes of their smallest elements. Define the (signed) weight of $(\pi,\rho)\in \mathcal{A}_{n,k}$ as $(-1)^{n-k}\alpha^{\nu_1(\rho)}\beta^{\nu_2(\rho)}x^j$, where $j$ denotes the number of non-special blocks of $\rho$. Then the left side of \eqref{eqnpaire2} is seen to give the sum of the weights of all members of $\mathcal{A}_n=\cup_{k=1}^n\mathcal{A}_{n,k}$, where we may assume $n \geq 1$.
Let $\mathcal{A}_n'$ denote the subset of $\mathcal{A}_n$ consisting of those $(\pi,\rho)$ such that $\pi=\{r+1\},\{r+2\},\ldots,\{r+n\}$ and within all blocks of $\rho$, the singletons of $\pi$ occur in descending order. To define an involution of $\mathcal{A}_n-\mathcal{A}_n'$, consider the largest $j\in I$, which we denote by $j^*$, such that either (i) the singleton $\{j\}$ occurs in $\pi$, with $\{j\}$ directly following another block $B$ of $\pi$ within some block of $\rho$ such that $j$ is greater than the largest element of $B$, or (ii) $j$ is the largest element of some non-singleton block of $\pi$. Replacing option (i) with (ii) by removing the singleton $\{j^*\}$ from $\pi$ and adding $j^*$ to the block $B$, or vice versa if (ii) occurs, is seen to define an involution of $\mathcal{A}_n-\mathcal{A}_n'$. It always preserves the weight (since the number of blocks of $\rho$ does not change and neither does its number of lr min), while it reverses the sign (the number of blocks of $\pi$ changing by one). Upon considering the number $k$ of singleton blocks of $\pi$ occurring in the non-special blocks of $\rho$, the sum of the weights of all members of $\mathcal{A}_n'$ is seen to be given by the right side of \eqref{eqnpaire2}, which completes the proof.
\end{proof}
The $L_r(n,k)$ serve as connection constants between the polynomial bases $\left((\alpha r +\beta x)^{\overline{n}}\right)_{n\geq0}$
and $\left(x^{\underline{n}}\right)_{n\geq 0}$, as pointed out in \cite[Corollary~4]{MS}. Here, we provide a combinatorial explanation as to why, which makes use of a sign-changing involution.
\begin{theorem}\label{conc}
If $n,r \geq 0$, then
\begin{equation}\label{conce1}
(\alpha r +\beta x)^{\overline{n}}=\sum_{j=0}^n L_r(n,j)x^{\underline{j}}.
\end{equation}
\end{theorem}
\begin{proof}
Given $\pi \in \mathcal{P}_{n,j}$, let $\pi^*$ denote a permutation of the non-special blocks of $\pi$. Let $m(\pi^*)$ be the number of left-right minima of $\pi^*$, where it is understood that the blocks permuted by $\pi^*$ (which are construed as the elements in a permutation) are ordered by the relative sizes of their respective smallest elements. Let $\widetilde{\mathcal{P}}_{n,j}=\{(\pi,\pi^*): \pi \in \mathcal{P}_{n,j}\}$, where it is understood that $\pi^*$ can range over all possible permutations of the non-special blocks of $\pi$ for each $\pi$. Given $\widetilde{\pi}=(\pi,\pi^*) \in \widetilde{\mathcal{P}}_{n,j}$, define the (signed) weight $w(\widetilde{\pi})$ by
$$ w(\widetilde{\pi})=(-1)^{j-m(\pi^*)}\alpha^{\nu_1(\pi)}\beta^{\nu_2(\pi)}x^{m(\pi^*)}.$$
It is seen from the definitions that the right-hand side of \eqref{conce1} gives the sum of the weights of all $\widetilde{\pi} \in \widetilde{\mathcal{P}}_n$, where $\widetilde{\mathcal{P}}_n=\cup_{j=0}^n \widetilde{\mathcal{P}}_{n,j}$.
To complete the proof, we define a sign-reversing involution on $\widetilde{\mathcal{P}}_n$ as follows. In each block permuted by $\pi^*$ within $\widetilde{\pi}=(\pi,\pi^*)\in \widetilde{\mathcal{P}}_{n,j}$, consider the ordering of the elements of $I$ that it contains. Decompose the ordering $s_\ell$ of the elements contained in the $\ell$-th block of $\pi^*$ (from left to right) by $$s_\ell=s_1^{(\ell)}\alpha_1^{(\ell)}s_2^{(\ell)}\alpha_2^{(\ell)}\cdots s_{r_{\ell}}^{(\ell)}\alpha_{r_{\ell}}^{(\ell)}, \qquad 1 \leq \ell \leq j,$$
where $s_1^{(\ell)}>s_2^{(\ell)}>\cdots>s_{r_{\ell}}^{(\ell)}$ denote the lr min of $s_\ell$ and the $\alpha_i^{(\ell)}$ are possibly empty. We call a sequence $s_\ell$ for some $\ell \in [j]$ \emph{disqualifying} if (i) $r_\ell \geq 2$ (i.e., block $\ell$ contains at least two lr min), or (ii) $r_\ell=1$ where $\ell>1$, with the $\ell$-th block of $\pi^*$ (when viewed from left to right) having larger first element than its predecessor.
Let $\ell'$ denote the largest $\ell$ such that $s_\ell$ is disqualifying. If $s_{\ell'}$ is disqualifying via (i), then break off the initial segment $s_1^{(\ell')}\alpha_1^{(\ell')}$ of $s_{\ell'}$ and form a separate (contents-ordered) block with it to directly follow the remaining part of the parent block. Otherwise, reverse this operation if (ii) applies. One may verify that this defines a sign-reversing involution on $\widetilde{\mathcal{P}}_n$ since $j$ always changes by one, and hence the sign, with the other factors in the definition of $w(\widetilde{\pi})$ unchanged. In particular, note that $m(\pi^*)$ does not change since blocks are ordered by the size of their smallest elements and thus an lr min is neither introduced nor removed when performing the operations above. The survivors of the involution are those members of $\widetilde{\mathcal{P}}_n$ in which the smallest element is first in each non-special block of $\pi$, with these blocks arranged in decreasing order of smallest elements from left to right (i.e., $\pi^*$ corresponds to the permutation $j(j-1)\cdots 1$). Each additional non-special block within such members of $\widetilde{\mathcal{P}}_n$ then yields a factor of $\beta x$ (as it corresponds to an lr min of $\pi^*$). Thus, the sum of the weights of the survivors is given by $(\alpha r +\beta x)(\alpha r+\beta x +1)\cdots(\alpha r+\beta x +n-1)=(\alpha r +\beta x)^{\overline{n}}$, since for each $i \in I$, one can either insert $i$ as an lr min in a special block, as a non-special block starter, or as a direct successor of some member of $[r+1,r+i-1]$.
\end{proof}
The following reciprocity result was shown in \cite{MS} using generating functions.
\begin{theorem}
If $n,r\geq0$, then
\begin{equation}\label{Lnrele1}
L_{n,r}^{(\alpha,\beta)}(x)=\sum_{k=0}^n (-1)^k\Lah{n}{k}L_{k,r}^{(-\alpha,-\beta)}(x).
\end{equation}
\end{theorem}
\begin{proof}
Let $\mathcal{U}_{n,k}$ denote the set of ordered pairs $\rho=(\rho_1,\rho_2)$, where $\rho_1$ is a partition of $[n]$ into $k$ contents-ordered blocks and $\rho_2 \in \mathcal{P}_k$, with the weight of $\rho$ taken to be that of $\rho_2$, but with $\alpha$ and $\beta$ replaced respectively by $-\alpha$ and $-\beta$. Let members of $\mathcal{U}_{n,k}$ have sign $(-1)^k$ and $\mathcal{U}_n=\cup_{k=0}^n\mathcal{U}_{n,k}$. Then the right side of \eqref{Lnrele1} gives the sum of the (signed) weights of all members of $\mathcal{U}_n$. To show that the left side of \eqref{Lnrele1} also achieves this, we first regard members $\rho$ of $\mathcal{U}_n$ as follows. Let $\rho'$ be obtained from $\rho$ by arranging the blocks of $\rho_1$, ordered by relative size of their smallest elements, according to the partition $\rho_2$ (where $r$ extra special elements are added that are distinct from the blocks of $\rho_1$). We let $\rho'$ have sign and weight equal to that of $\rho$. Thus, the right side of \eqref{Lnrele1} gives the sum of the weights of all configurations $\rho'$, the set of which we again denote by $\mathcal{U}_n$.
We define an involution on $\mathcal{U}_n$ as follows. Let us refer to the blocks of $\rho'$ whose elements are themselves blocks of $\rho_1$ as \emph{superblocks}. In a left-to-right scan of the superblocks of $\rho' \in \mathcal{U}_n$, where the special superblocks are written before the non-special, consider the first, denoted by $\mathcal{B}$, which contains within its blocks at least two elements of $[n]$ altogether. Let $B$ denote the rightmost block within $\mathcal{B}$. If $B$ is not a singleton, then break off the final element of $B$ and form a singleton block with it to follow $B$. If $B$ is a singleton, then we add its element to the block that directly precedes it at the end. Note that this operation always reverses the sign and preserves the weight except in the cases (I) $B=\{\alpha b\beta a\}$ or (II) $B'=\{\alpha b\beta\}$ and $B=\{a\}$, where $B'$ denotes the penultimate block of $\mathcal{B}$, $a**0$ for $1 \leq k \leq n$. Then $\mathbf{M}_r(n,k)^2 \geq \mathbf{M}_r(n,k-1)\mathbf{M}_r(n,k+1)$ if and only if
\begin{align}
&\big(\beta\mathbf{M}_r(n-1,k-1)+(\alpha r_p+\beta k_q+n-1)\mathbf{M}_r(n-1,k)\big)^2\notag\\
&\quad\geq \big(\beta\mathbf{M}_r(n-1,k-2)+(\alpha r_p+\beta (k-1)_q
+n-1)\mathbf{M}_r(n-1,k-1)\big)\cdot\big(\beta\mathbf{M}_r(n-1,k)\notag\\
&\quad\quad+(\alpha r_p+\beta (k+1)_q+n-1)\mathbf{M}_r(n-1,k+1)\big).\label{logcone1}
\end{align}
Upon expanding both sides and comparing the various terms, it suffices to show
\small\begin{equation}\label{logcone2}
\left(\alpha r_p+\beta k_q+n-1\right)^2 \geq \left(\alpha r_p+\beta (k-1)_q+ n-1\right)\left(\alpha r_p +\beta (k+1)_q+n-1\right)
\end{equation}\normalsize
and
\small\begin{align}
&\left(\alpha r_p+\beta(2k_q-(k-1)_q)+n-1\right)\mathbf{M}_r(n-1,k-1)\mathbf{M}_r(n-1,k) \notag\\
&\qquad\qquad\qquad\geq \left(\alpha r_p+\beta(k+1)_q+n-1\right)\textbf{M}_r(n-1,k-2)\textbf{M}_r(n-1,k+1)\label{logcone3},
\end{align}\normalsize
from which \eqref{logcone1} would follow by induction. Note that \eqref{logcone2} may be simplified to
$$2(\alpha r_p+n-1)k_q+\beta k_q^2 \geq (\alpha r_p+n-1)((k-1)_q+(k+1)_q)+\beta(k-1)_q(k+1)_q.$$
The last inequality follows from observing that $2k_q \geq (k-1)_q+(k+1)_q$ for $0 \leq q \leq 1$ and $k_q^2\geq (k-1)_q(k+1)_q$ for all $ q \geq 0$. Inequality \eqref{logcone3} follows from observing that $\mathbf{M}_r(n-1,k-1)\mathbf{M}_r(n-1,k) \geq \mathbf{M}_r(n-1,k-2)\mathbf{M}_r(n-1,k+1)$ since the ratio $\frac{\mathbf{M}_r(n-1,\ell)}{\mathbf{M}_r(n-1,\ell+1)}$ is increasing by the induction hypothesis and since $2k_q-(k-1)_q\geq(k+1)_q$, which completes the proof.
\end{proof}
\begin{remark} The $p=q=r=1$ case of Theorem \ref{logcon} is equivalent to \cite[Corollary~13]{MS}, which was shown by a different method making use of Newton's inequality.
\end{remark}
Extending prior proofs yields generalizations of several of the identities given above for $L_r(n,k)$. For example, extending the combinatorial proof of \eqref{neqnse1} to account for the $\sigma_1$ and $\sigma_2$ statistics gives
\begin{equation}\label{neqnse1ext}
\mathbf{L}_r(n+m,k)=\sum_{i=0}^n\sum_{j=0}^m\binom{n}{i}\binom{m}{j}\left(\alpha p^s(r-s)_p\right)^{\overline{i+j}}\mathbf{L}_{s}(n+m-i-j,k), \qquad 0 \leq s \leq r,
\end{equation}
the $m=0$ case of which is
\begin{equation}\label{furecse1ext}
\mathbf{L}_r(n+m,k)=\sum_{j=k}^n\binom{n}{j}(\alpha p^s(r-s)_p)^{\overline{n-j}}\mathbf{L}_s(j,k).
\end{equation}
Note that \eqref{furecse1ext} may also be shown by an algebraic argument comparable to the one given above for \eqref{furecse1}, using \eqref{genexpforme1} in place of \eqref{exforme1}.
The proof given above for \eqref{L(n,k)rece2} may be generalized to give
\begin{align}
\mathbf{L}_r^{(\alpha,\beta)}(n+1,m+j+1;p,q)=\beta q^m\sum_{k=m}^n\sum_{\ell=j}^{n-k}&\binom{n-k}{\ell}\left(\alpha r_p+\beta (m+1)_q+k+1\right)^{\overline{n-k-\ell}}\notag\\
&\times \mathbf{L}_r^{(\alpha,\beta)}(k,m;p,q)\mathbf{L}_0^{(\alpha,\beta q^{m+1})}(\ell,j;p,q).\label{L(n,k)rece2ext}
\end{align}
Note that the $q^m$ factor at the beginning accounts for the lr min created when $m+1$ is initially placed in the $(k+1)$-st non-special block. Also, the $\beta$ parameter must be replaced by $\beta q^{m+1}$ in the final factor within the summand since each lr min within the final $j$ non-special blocks contributes an extra $m+1$ (more than it ordinarily would) towards the $\sigma_2$ statistic value.
Finally, extending prior arguments yields the following recurrence for $\mathbf{L}_{n,r}(x)$:
\begin{equation}\label{mathbfPrec}
\mathbf{L}_{m+n,r}^{(\alpha,\beta)}(x;p,q)=\sum_{i=0}^m\sum_{j=0}^n\binom{n}{j}x^i\left(\alpha p^{r-s}s_p+\beta i_q+m\right)^{\overline{n-j}}\mathbf{L}_r^{(\alpha,\beta)}(m,i;p,q)\mathbf{L}_{j,r-s}^{(\alpha,\beta q^i)}(x;p,q),
\end{equation}
which provides a ($p,q$)-analogue of \eqref{P{n,r}forme1}.
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\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}: Primary 05A19; Secondary 05A05.
\noindent \emph{Keywords:} Laguerre polynomial, $q$-generalization, partial $r$-Bell polynomial, combinatorial proof.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A000110},
\seqnum{A000262},
\seqnum{A008275},
\seqnum{A008277}, and
\seqnum{A008297}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received November 2 2020;
revised version received January 16 2021.
Published in {\it Journal of Integer Sequences} January 23 2021.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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\end{document}
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