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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…
It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by…
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent…
In this paper we present grammatical interpretations of the alternating Eulerian polynomials of types A and B. As applications, we derive several properties of the type B alternating Eulerian polynomials, including combinatorial expansions,…
In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.
In this note, by using the Hasse-Teichm\"uller derivatives, we obtain two explicit expressions for the related numbers of higher order Appell polynomials. One of them presents a determinant expression for the related numbers of higher order…
The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$…
In this paper, applying the Fa\`a di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials…
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties.…
In this paper, we first present combinatorial proofs of a kind of expansions of the Eulerian polynomials of types A and B, and then we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the…
The study of degenerate versions of certain special polynomials and numbers, which was initiated by Carlitz's work on degenerate Euler and degenerate Bernoulli polynomials, has recently seen renewed interest among mathematicians. The aim of…
The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.
We introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Then, among other things, by using the formula about representing one lambda-Sheffer polynomial in terms of other lambda-Sheffer…
Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the…
In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these…
We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and…
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…
In this paper, we choose the derivative polynomials for tangent and secant as basis sets of polynomial space. From this viewpoint, we first give an expansion of the derivative polynomials for tangent in terms of the derivative polynomials…