Restricted Stirling and Lah number matrices and their inverses

TL;DR

This paper studies restricted Stirling and Lah number matrices, providing explicit combinatorial formulas for their inverses and answering longstanding questions about their interpretations.

Contribution

It introduces combinatorial interpretations for the inverses of restricted Stirling and Lah number matrices, extending classical results and addressing open questions.

Findings

Explicit formulas for inverse matrices entries as differences of labeled forests.

Combinatorial interpretations for inverses of restricted Stirling and Lah matrices.

Connections to Whitney numbers of certain partition posets.

Abstract

Given $R \subseteq \mathbb{N}$ let ${n \brace k}_R$, ${n \brack k}_R$, and $L(n,k)_R$ be the number of ways of partitioning the set $[n]$ into $k$ non-empty subsets, cycles and lists, respectively, with each block having cardinality in $R$. We refer to these as the $R$-restricted Stirling numbers of the second and first kind and the $R$-restricted Lah numbers, respectively. Note that the classical Stirling numbers of the second kind and first kind, and Lah numbers are ${n \brace k} = {n \brace k}_{\mathbb{N}}$, ${n \brack k} = {n \brack k}_{\mathbb{N}} $ and $L(n,k) = L(n,k)_{\mathbb{N}}$, respectively. The matrices $[{n \brace k}]_{n,k \geq 1}$, $[{n \brack k}]_{n,k \geq 1}$ and $[L(n,k)]_{n,k \geq 1}$ have inverses $[(-1)^{n-k}{n \brack k}]_{n,k \geq 1}$, $[(-1)^{n-k} {n \brace k}]_{n,k \geq 1}$ and $[(-1)^{n-k} L(n,k)]_{n,k \geq 1}$ respectively. The inverse matrices $[{n \brace k}_R]^{-1}_{n,k \geq 1}$, $[{n \brack k}_R]^{-1}_{n,k \geq 1}$ and $[L(n,k)_R]^{-1}_{n,k \geq 1}$ exist if and only if $1 \in R$. We express each entry of each of these matrices as the difference between the cardinalities of two explicitly defined families of labeled forests. In particular the entries of $[{n \brace k}_{[r]}]^{-1}_{n,k \geq 1}$ have combinatorial interpretations, affirmatively answering a question of Choi, Long, Ng and Smith from 2006. If $1,2 \in R$ and if for all $n \in R$ with $n$ odd and $n \geq 3$, we have $n \pm 1 \in R$, we additionally show that each entry of $[{n \brace k}_R]^{-1}_{n,k \geq 1}$, $[{n \brack k}_R]^{-1}_{n,k \geq 1}$ and $[L(n,k)_R]^{-1}_{n,k \geq 1}$ is up to an explicit sign the cardinality of a single explicitly defined family of labeled forests. Our results also provide combinatorial interpretations of the $k$th Whitney numbers of the first and second kinds of $\Pi_n^{1,d}$, the poset of partitions of $[n]$ that have each part size congruent to $1$ mod $d$.