  It is observed that the conjugacy growth series of the infinite fini-tary
symmetric group with respect to the generating set of transpositions is the
generating series of the partition function. Other conjugacy growth series are
computed, for other generating sets, for restricted permutational wreath
products of finite groups by the finitary symmetric group, and for alternating
groups. Similar methods are used to compute usual growth polynomials and
conjugacy growth polynomials for finite symmetric groups and alternating
groups, with respect to various generating sets of transpositions. Computations
suggest a class of finite graphs, that we call partition-complete, which
generalizes the class of semi-hamiltonian graphs, and which is of independent
interest. The coefficients of a series related to the finitary alternating
group satisfy congruence relations analogous to Ramanujan congruences for the
partition function. They follow from partly conjectural "generalized Ramanujan
congruences", as we call them, for which we give numerical evidence in Appendix
C.
