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.

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

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 André permutations.

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.