Derivative polynomials and enumeration of permutations by number of interior and left peaks

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 number of interior and left peaks over the symmetric group. Properties of the generating functions for the number of interior and left peaks over the symmetric group, including recurrence relations, generating functions and real-rootedness, are studied.