Limit Densities of Patterns in Permutation Inflations

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