Limit Densities Of Patterns In Permutation Inflations

Call a permutation k-inflatable if the sequence of its tensor products with uniform random permutations of increasing lengths has uniform k-point pattern densities. Previous work has shown that nontrivial k-inflatable permutations do not exist for k >= 4. In this paper, we derive a general formula for the limit densities of patterns in the sequence of tensor products of a fixed permutation with each permutation from a convergent sequence. By applying this result, we completely characterize 3-inflatable permutations and find explicit examples of 3-inflatable permutations with various lengths, including the shortest examples with length 17.