Restricted Stirling and Lah numbers, and their inverses

Abstract

Let ${n \brace k}_{[r]}$ denote the number of ways of partitioning a set of size $n$ into $k$ non-empty blocks, each of which has size at most $r$. For all $r$ we find a combinatorial interpretation for the entries of the inverse of the matrix $\left[{n \brace k}_{[r]}\right]_{n,k \geq 1}$. For even $r$ we exhibit sets of forests counted by the entries of the inverse. For odd $r$ our interpretation is as the difference in size between two sets of forests. This answers a question raised by Choi, Long, Ng and Smith in 2006. More generally we consider restricted Stirling numbers of the second and first kinds ${n \brace k}_{R}$, ${n \brack k}_{R}$, and Lah numbers $L(n,k)_{R}$, for ${R} \subseteq {\mathbb N}$. These are defined to be the number of ways of partitioning a set of size $n$ into $k$ non-empty blocks (for Stirling numbers of the second kind), cycles (for Stirling numbers of the first kind) or lists (for Lah numbers) with the size of each block, cycle or list in ${R}$. For any ${R}$ satisfying $1 \in {R}$ (a necessary condition for the inverses to exist) we find combinatorial interpretations for the entries of the inverses of the matrices $\left[{n \brace k}_{R}\right]_{n,k \geq 1}$, $\left[{n \brack k}_{R}\right]_{n,k \geq 1}$ and $\left[L(n,k)_{R}\right]_{n,k \geq 1}$, as the difference in size between two sets of forests. In the case of Stirling numbers of the second kind and Lah numbers, for certain ${R}$ we can do better, interpreting the inverse entries directly as counts of single sets of forests. Among these ${R}$'s are those which include $1$ and $2$ and which have the property that for all odd $n \in {R}$, $n \geq 3$, we have $n\pm1 \in {R}$. Our proofs depend in part on two combinatorial interpretations of the coefficients of the reversion (compositional inverse) of a power series.