Unimodal sequences and quantum and mock modular forms

摘要

We show that the rank generating function U(t;q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U(-1; q) to obtain a quantum-modular form which is "dual" to the quantum form Zagier constructed from Kontsevich's "strange" function F(q). As a result, we obtain a new representation for a certain generating function for L-values. The series U(i;q) = U(-i;q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.