  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 $p(n)_{\textbf{e}}$. 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 $p(n)_{\textbf{e}}$ and prove their
conjectured congruences.
