Q-Enumeration of alternating permutations of odd length

Abstract

A permutation π = π1 π2 … πn is called alternating (up-down ) permutation if π < π > π < π > π…. For odd n, their number is given by n![zn ] tan z, and for even n by n![zn ] sec z. It was shown in [5] that the number of alternating permutations of odd length with i inversions is given by and the number of alternating permutations of even length with i inversions is given by . In [7], several q-analogues of both secant and tangent numbers are given. In [4], Cristea and Prodinger consider the q-enumeration of up-down words by number of rises. In this project we give complete proofs of the q-analogues of the tangent numbers using the method explained in [7].