KAIST Discrete Math Seminar

The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(-q)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function φ(q). Congruences for the smallest parts functions associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler’s pentagonal theorem are also obtained. This is joint work with George Andrews and Atul Dixit.

The Euler number E_n counts the number of alternating permutations on the set [n]. It is well known that its exponential generating function equals Tan z + Sec z. For this reason, E_2n and E_2n+1 are called secant numbers and tangent numbers, respectively. Certain polynomials arising in series expansions for zeros of generalized Rogers-Ramanujan functions provide a q-analog of the tangent numbers, which is part of a wider class of polynomials with similar combinatorial interpretations. In this talk, we will discuss various q-Euler numbers. This is a joint work with Tim Huber from Iowa State University.