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

Associated rr-Dowling numbers and some relatives

In this paper, we introduce a new generalization of Bell numbers, the $s$-associated $r$-Dowling numbers by combining two investigational directions. Here, $r$ distinguished elements have to be in distinct blocks, some elements are coloured according to a colouring rule, and the cardinality of certain blocks is bounded from below by $s$. Along with them, we define some relatives, the $s$-associated $r$-Dowling factorials and the $s$-associated $r$-Dowling–Lah numbers, when the underlying set is decomposed into cycles or ordered blocks. The study of these numbers is based on their combinatorial meaning, and the exponential generating functions of their sequences derived from the so-called $r$-compositional formula

Associated rr-Dowling numbers and some relatives

In this paper, we introduce a new generalization of Bell numbers, the $s$-associated $r$-Dowling numbers by combining two investigational directions. Here, $r$ distinguished elements have to be in distinct blocks, some elements are coloured according to a colouring rule, and the cardinality of certain blocks is bounded from below by $s$. Along with them, we define some relatives, the $s$-associated $r$-Dowling factorials and the $s$-associated $r$-Dowling–Lah numbers, when the underlying set is decomposed into cycles or ordered blocks. The study of these numbers is based on their combinatorial meaning, and the exponential generating functions of their sequences derived from the so-called $r$-compositional formula

Translated Whitney Numbers and Their q-Analogues

Abstract This paper presents natural q-analogues for the translated Whitney numbers. Several combinatorial properties which appear to be q-deformations of those classical ones are obtained. Moreover, we give a combinatorial interpretation of the classical translated Whitney numbers of the first and second kind, and their qanalogues in terms of A-tableaux

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials'$ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial$ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya$ and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure

New combinatorial interpretations of r-Whitney and r-Whitney-Lah numbers

T. A. Dowling introduced Whitney numbers of the first and second kind concerning the so-called Dowling lattices of finite groups. It turned out that they are generalizations of Stirling numbers. Later, I. Mező defined r-Whitney numbers as common generalizations of Whitney numbers and r-Stirling numbers. Additionally, G.-S. Cheon and J.-H. Jung defined r-Whitney-Lah numbers. In our paper, we give new combinatorial interpretations of r-Whitney and r-Whitney-Lah numbers, which correspond better with the combinatorial definitions of Stirling, r-Stirling, Lah and r-Lah numbers. These allow us to explain their properties in a purely combinatorial manner, as well as derive several new identities

Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers

Ordering identities in the Weyl-Heisenberg algebra generated by single-mode boson operators are investigated. A boson string composed of creation and annihilation operators can be expanded as a linear combination of other such strings, the simplest example being a normal ordering. The case when each string contains only one annihilation operator is already combinatorially nontrivial. Two kinds of expansion are derived: (i) that of a power of a string $\Omega$ in lower powers of another string $\Omega'$, and (ii) that of a power of $\Omega$ in twisted versions of the same power of $\Omega'$. The expansion coefficients are shown to be, respectively, the generalized Stirling numbers of Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are given. These combinatorial numbers are binomial transforms of each other, and their theory is developed, emphasizing schemes for computing them: summation formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating hypergeometric series, and closed-form expressions. The results on the first type of expansion subsume a number of previous results on the normal ordering of boson strings.Comment: 36 pages (preprint format