Related papers: A note on Stirling permutations
In 1977 Carlitz and Scoville introduced the cycle $(\alpha,t)$-Eulerian polynomials $A^{\mathrm{cyc}}_n(x,y, t\,|\,\alpha)$ by enumerating permutations with respect to the number of excedances, drops, fixed points and cycles. In this paper,…
A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of…
In the present article we introduce two new combinatorial interpretations of the $r$-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar $G:=\{ y\rightarrow yx^{m},…
We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove…
In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $\gamma_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $\pi^{-2}$ with rational…
In this work we propose a combinatorial model that generalizes the standard definition of permutation. Our model generalizes the degenerate Eulerian polynomials and numbers of Carlitz from 1979 and provides missing combinatorial proofs for…
In this paper, algorithms are developed for computing the Stirling transform and the inverse Stirling transform; specifically, we investigate a class of sequences satisfying a two-term recurrence. We derive a general identity which…
We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Br\"and\'en. Additionally, we find $q$-exponential…
We derive two new identities involving the Bernoulli numbers, the Euler numbers, and the Stirling numbers of the first kind using analytic continuation of a well known identity for the Stirling numbers of the first kind.
A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on $\mathfrak{S}_n$ was given by D\'{e}sarm\'{e}nien and Foata in 1992, and a refined version, called signed Euler-Mahonian identity, together with a…
In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family…
As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given…
The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernolli, degenerate…
We introduce multinomial and $r$-variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
The $(q,r)$-Eulerian polynomials are the $(\maj-$$\exc,\fix,\exc)$ enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical…
Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations, where a Cayley permutation is a word of positive integers such that if…
We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We…
In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…
We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and…