Theoretical analysis reveals the limiting mean of longest increasing subsequences scales as 4\sqrt{nk} in wreath product permutations, indicating distinct behavior from colored permutations.
Let Gn,k be the group of permutations of \{1,2,…, 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 σ chosen uniformly in Gn,k , let Ln,k be the length of the longest increasing subsequence in σ . For k , n growing, we determine that the limiting mean of Ln,k is asymptotic to 4√nk . This is different from parallel variations of the Vershik–Kerov theorem for colored permutations.
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Chatterjee et al. (2026) studied this question.
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