The Jacobi-Stirling Numbers

The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which, as shown in LW, are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations thereby extending and supplementing known contributions to the literature of Andrews-Littlejohn, Andrews-Gawronski-Littlejohn, Egge, Gelineau-Zeng, and Mongelli.Comment: 17 pages, 3 table

Generator Sets for the Alternating Group

Although the alternating group is an index 2 subgroup of the symmetric group, there is no generating set that gives a Coxeter structure on it. Various generating sets were suggested and studied by Bourbaki, Mitsuhashi, Regev-Roichman, Vershik-Vserminov and others. In a recent work of Brenti- Reiner-Roichman it is explained that palindromes in Mitsuhashi's generating set play a role similar to that of re ections in a Coxeter system. We study in detail the length function with respect to the set of palindromes. Results include an explicit combinatorial description, a generating function, and an interesting connection to Broder's restricted Stirling numbers

On umbral extensions of Stirling numbers and Dobinski-like formulas

Umbral extensions of the stirling numbers of the second kind are considered and the resulting dobinski-like various formulas including new ones are presented. These extensions naturally encompass the two well known q-extensions. The further consecutive umbral extensions q-stirling numbers are therefore realized here in a two-fold way. The fact that the umbral q-extended dobinski formula may also be interpreted as the average of powers of random variable with the q-poisson distribution singles out the q-extensions which appear to be a kind of singular point in the domain of umbral extensions as expressed by corresponding two observations. Other relevant possibilities are tackled with the paper`s closing down questions and suggestions with respect to other already existing extensions while a brief limited survey of these other type extensions is being delivered. There the newton interpolation formula and divided differences appear helpful and inevitable along with umbra symbolic language in describing properties of general exponential polynomials of touchard and their possible generalizations. Exponential structures or algebraically equivalent prefabs with their exponential formula appear to be also naturally relevant.Comment: 40 page

Associated Lah numbers and r-Stirling numbers

We introduce the associated Lah numbers. Some recurrence relations and convolution identities are established. An extension of the associated Stirling and Lah numbers to the r-Stirling and r-Lah numbers are also given. For all these sequences we give combinatorial interpretation, generating functions, recurrence relations, convolution identities. In the sequel, we develop a section on nested sums related to binomial coefficient

Colorful Proofs of the Generating Formulas for Signed and Unsigned Stirling Numbers of the First Kind

We describe proofs of the standard generating formulas for unsigned and signed Stirling numbers of the first kind that follow from a natural combinatorial interpretation based on cycle-colored permutations.Comment: 5 pages, 2 figure

New families of special numbers for computing negative order Euler numbers

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem \cite{golombek} \textquotedblleft Aufgabe 1088, El. Math. 49 (1994) 126-127\textquotedblright . Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by some tables. We give some applications in Probability and Statistics. That is, special values of mathematical expectation in the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we give two algorithms for computation our numbers. We also give some combinatorial applications, further remarks on our numbers and their generating functions.Comment: arXiv admin note: text overlap with arXiv:1604.0560

Analogies of the Qi formula for some Dowling type numbers

In the paper, the authors establish explicit formulas for the Dowling numbers and their generalizations in terms of generalizations of the Lah numbers and the Stirling numbers of the second kind. These results gen- eralize the Qi formula for the Bell numbers in terms of the Lah numbers and the Stirling numbers of the second kind

A note on degenerate Stirling numbers of the first kind

Recently, the degenerate Stirling numbers of the first kind were introduced. In this paper, we give some formulas for the degenerate Stirling numbers of the first kind in the terms of the complete Bell polynomials with higher-order harmonic number arguments. Also, we derive an identity connecting the degenerate Stirling numbers of the first kind and the degenerate derangement numbers by using probabilistic method.Comment: 12 page

New formulas for Stirling-like numbers and Dobinski-like formulas

Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $[1]$. These extensions naturally encompass the well known $q$- extensions. The indicatory references are to point at a part of the vast domain of the foundations of computer science in arxiv affiliation.Comment: 9 pages, presented at the Gian-Carlo Rota Polish Seminar, http://ii.uwb.edu.pl/akk/sem/sem_rota.ht

q-Stirling numbers: A new view

We show the classical $q$-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative result via a decomposition of a new poset $\Pi(n,k)$ which we call the Stirling poset of the second kind. Its rank generating function is the $q$-Stirling number $S_q[n,k]$. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done. Letting $t = 1+q$ we give a bijective argument showing the $(q,t)$-Stirling numbers of the first and second kind are orthogonal