Restricted Stirling and Lah number matrices and their inverses

Abstract

Given R⊆N let {nk}R, [nk]R, and L(n,k)R count the number of ways of partitioning the set [n]:={1,2,…,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 kind, R-restricted unsigned Stirling numbers of the first kind and the R-restricted Lah numbers, respectively. Note that the classical Stirling numbers of the second kind, unsigned Stirling numbers of the first kind, and Lah numbers are {nk}={nk}N, [nk]=[nk]N and L(n,k)=L(n,k)N, respectively.It is well-known that the infinite matrices [{nk}]n,k≥1, [[nk]]n,k≥1 and [L(n,k)]n,k≥1 have inverses [(−1)n−k[nk]]n,k≥1, [(−1)n−k{nk}]n,k≥1 and [(−1)n−kL(n,k)]n,k≥1 respectively. The inverse matrices [{nk}R]n,k≥1−1, [[nk]R]n,k≥1−1 and [L(n,k)R]−1n,k≥1 exist and have integer entries if and only if 1∈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 [{nk}[r]]−1n,k≥1 have combinatorial interpretations, affirmatively answering a question of Choi, Long, Ng and Smith from 2006.If we have 1,2∈R and if for all n∈R with n odd and n≥3, we have n±1∈R, we additionally show that each entry of [{nk}R]n,k≥1−1, [[nk]R]−1n,k≥1 and [L(n,k)R]n,k≥1−1 is up to an explicit sign the cardinality of a single explicitly defined family of labeled forests. With R as before we also do the same for restriction sets of the form R(d)={d(r−1)+1:r∈R} for all d≥1. Our results also provide combinatorial interpretations of the kth Whitney numbers of the first and second kinds of Πn1,d, the poset of partitions of [n] that have each part size congruent to 1 mod d.