Restricted Stirling and Lah Number Matrices and Their Inverses

Given R⊆Nlet {nk}R, [nk]R, and L(n, k)Rcount the number of ways of partitioning the set [n] :={1, 2, ..., n}into knon-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]Nand L(n, k) =L(n, k)N, respectively.It is well-known that the infinite matrices [{nk}]n,k≥1, [[nk]]n,k≥1and [L(n, k)]n,k≥1have inverses [(−1)n−k[nk]]n,k≥1,[(−1)n−k{nk}]n,k≥1and [(−1)n−kL(n, k)]n,k≥1respectively. The inverse matrices [{nk}R]−1n,k≥1, [[nk]R]−1n,k≥1and[L(n, k)R]−1n,k≥1exist 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≥1have combinatorial interpretations, affirmatively answering a question of Choi, Long, Ng and Smith from 2006.If we have 1, 2 ∈Rand if for all n ∈Rwith nodd and n ≥3, we have n ±1 ∈R, we additionally show that each entry of [{nk}R]−1n,k≥1, [[nk]R]−1n,k≥1and [L(n, k)R]−1n,k≥1is up to an explicit sign the cardinality of a single explicitly defined family of labeled forests. With Ras 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 Π1,dn, the poset of partitions of [n]that have each part size congruent to 1mod d

Elliptic rook and file numbers

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of Garsia and Remmel's q-file numbers for skyline boards. We also provide an elliptic extension of the j-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings has been taken out to keep the length of this paper reasonable

Combinatorially interpreting generalized Stirling numbers

Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any smooth function $f(x)$, defines a sequence $(S_w(k))_k$, the terms of which we refer to as the {\em Stirling numbers (of the second kind)} of $w$. The nomenclature comes from the fact that when $w=(xD)^n$, we have $S_w(k)={n \brace k}$, the ordinary Stirling number of the second kind. Explicit expressions for, and identities satisfied by, the $S_w(k)$ have been obtained by numerous authors, and combinatorial interpretations have been presented. Here we provide a new combinatorial interpretation that retains the spirit of the familiar interpretation of ${n \brace k}$ as a count of partitions. Specifically, we associate to each $w$ a quasi-threshold graph $G_w$, and we show that $S_w(k)$ enumerates partitions of the vertex set of $G_w$ into classes that do not span an edge of $G_w$. We also discuss some relatives of, and consequences of, our interpretation, including $q$-analogs and bijections between families of labelled forests and sets of restricted partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00