Abstract:
Let R(n,k) denote the number of permutations of {1,2,...,n} with k alternating runs. We find a grammatical description of the numbers R(n,k) and then present several convolution formulas involving the generating function for the numbers R(n,k). Moreover, we establish a connection between alternating runs and Andre permutations.

Abstract:
Let $R(n,k)$ denote the number of permutations of ${1,2,...,n}$ with $k$ alternating runs. In this note we present an explicit formula for the numbers $R(n,k)$.

Abstract:
The purpose of this paper is to show that some combinatorial sequences, such as second-order Eulerian numbers and Eulerian numbers of type $B$, can be generated by context-free grammars.

Abstract:
The purpose of this paper is to show that Bessel polynomials, double factorials and Catalan triangle can be generated by using context-free grammars.

Abstract:
In this paper we consider the gamma-vectors of the types A and B Coxeter complexes as well as the gamma-vectors of the types A and B associahedrons. We show that these gamma-vectors can be obtained by using derivative polynomials of the tangent and secant functions. A grammatical description for these gamma-vectors is discussed. Moreover, we also present a grammatical description for the well known Legendre polynomials and Chebyshev polynomials of both kinds.

Abstract:
The purpose of this paper is to study some binomial coefficients which are related to the evaluation of tan(nx). We present a connection between these binomial coefficients and the coefficients of a family of derivative polynomials for tangent and secant.

Abstract:
A remarkable identity involving the Eulerian polynomials of type D was obtained by Stembridge (Adv. Math. 106 (1994), p. 280, Lemma 9.1). In this paper we explore an equivalent form of this identity. We prove Brenti's real-rootedness conjecture for the Eulerian polynomials of type $D$.

Abstract:
In this paper we introduce a family of two-variable derivative polynomials for tangent and secant. We study the generating functions for the coefficients of this family of polynomials. In particular, we establish a connection between these generating functions and Eulerian polynomials.

Abstract:
Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left peaks over the symmetric group. Properties of the generating functions for the numbers of interior and left peaks over the symmetric group, including recurrence relations, generating functions and real-rootedness, are studied.

Abstract:
In this paper, we establish a connection between the 1/k-Eulerian polynomials introduced by Savage and Viswanathan (Electron. J. Combin. 19(2012), P9) and k-Stirling permutations. We also introduce the dual set of Stirling permutations.