Restricted Stirling and Lah number matrices and their inverses

Abstract

Given R ⊆ N let n k R , n k 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 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 k = n k N , n k = n k N and L(n, k) = L(n, k) N , respectively.n,k≥1 and [L(n, k) R ] −1 n,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 [ n k [r] ] −1 n,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 [ n k R ] −1 n,k≥1 , [ n k R ] −1 n,k≥1 and [L(n, k) R ] −1 n,k≥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 Π 1,d n , the poset of partitions of [n] that have each part size congruent to 1 mod d.

Graph rules for the linked cluster expansion of the Legendre effective action

“…The result (B5) then follows from (B6). The formula (B6) has several known combinatorial and tree graph interpretations, see 10,11,14 and the references therein. In the remainder of this appendix we present a graph theoretical interpretation of (B4), (B5) which, together with its proof, mirrors some aspects of its quantum field theoretical counterpart in Section III.…”

“…By the results of Section IV the subsums over vertex labeled trees T ∈ T (B(v), n) with fixed weight µ(T ) have a combinatorial meaning in terms of the number of integer labeled tree graphs of the same topology as T . The graph rule could therefore optimized once explicit results for the number of set partitions P(D(v), D n ) are available; see 10 for some related results.…”

“…A convenient generating function is exp{y(e x −1)} = ∑ I,n≥0 S(I, n) y n x I /I!. Generalizations have been considered in 10 .…”

Restricted r-Stirling numbers and their combinatorial applications

“…}, we recover the associated r-Stirling numbers of the second kind [19]. Moreover, if r = 0 we have the S-restricted Stirling numbers of the second kind [6,12,29].…”

“…It is clear that the row sum of the matrix M S,r are the (S, r)-Bell numbers B n,S,r . The inverse exponential Riordan array of M S,r and L S,r are denoted byT S,r := n k −,r n,k≥0and U S,r := n k −,r n,k≥0.For the particular case r = 0, Engbers et al[12] gave an interesting combinatorial interpretation for the absolute values of the entries n k −by using Schröder trees. Since M S,r * T S,r = I, where I is the identity matrix, we have the orthogonality relation: r = δ k,n .…”