A Vershik–Kerov theorem for wreath products

Let G_{n,k} be the group of permutations of \{1,2,\ldots, kn\} that permutes the first k symbols arbitrarily, then the next k symbols and so on through the last k symbols. Finally, the n blocks of size k are permuted in an arbitrary way. For \sigma chosen uniformly in G_{n,k} , let L_{n,k} be the length of the longest increasing subsequence in \sigma . For k , n growing, we determine that the limiting mean of L_{n,k} is asymptotic to 4\sqrt{nk} . This is different from parallel variations of the Vershik–Kerov theorem for colored permutations.