Asymptotics and congruences for partition functions which arise from finitary permutation groups

In a recent paper, Bacher and de la Harpe study the conjugacy growth series of finitary permutation groups. In the course of studying the coefficients of a series related to the finitary alternating group, they introduce generalized partition functions \(p(n)_\mathbf{e }\). The group theory motivates the study of the asymptotics for these functions. Moreover, Bacher and de la Harpe also conjecture over 200 congruences for these functions which are analogous to the Ramanujan congruences for the unrestricted partition function p(n). We obtain asymptotic formulas for all of the \(p(n)_\mathbf{e }\) and prove their conjectured congruences.