We examine the conjugacy growth series of all wreath products of the finitary permutation groups $\text{ Sym}(X)$ and $\text{ Alt}(X)$ for an infinite set $X$. We determine their asymptotics, and we characterize the limiting behavior between the $\text{ Alt}(X)$ and $\text{ Sym}(X)$ wreath products. In particular, their ratios form a limit if and only if the dimension of the symmetric wreath product is twice the dimension of the alternating wreath product.
Categories: math.NT math.CO math.GR