Combinatorics of geometrically distributed random variables: new q-tangent and q-secant numbers

Abstract

Up-down permutations are counted by tangent (respectively, secant) numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all coincide with the classical version. In this way, we get some new q-tangent and q-secant functions. Some of them also have nice continued fraction expansions; in one particular case, we could not find a proof for it. Divisibility results à la Andrews, Foata, Gessel are also discussed.

Keywords

• Tangent numbers
• secant numbers
• alternating permutations
• geometric distribution
• q-analogues.