The average number of cycles. (Q1042422)

The cycle number indicator for a permutation group \(G\) of degree \(n\) is \(z_G(t):=\tfrac 1{|G|}\sum_{k=1}^n\nu_G(k)t^k\) where \(\nu_G(k)\) is the number of permutations in \(G\) with \(k\) cycles. It is easily seen that the average number \(e_G\) of cycles is \(z_G'(1)\) and the variance \(\sigma_G^2\) is \(z_G''(1)+z_G'(1)-z_G'(1)^2\). The values \(e_G\) and \(\sigma_G^2\) for \(G=S_n\) are well known, and the authors show that the corresponding values for \(G=A_n\) are asymptotic to these. For the (imprimitive) wreath product \(W:=G\wr H\) we have \(z_W=z_H(z_G(t))\) and so \(e_W=e_Ge_H\) and \(\sigma_W^2=e_G^2\sigma_H^2+\sigma_G^2e_H\).