Limit Densities of Patterns in Permutation Inflations

TL;DR

This paper characterizes 3-inflatable permutations by deriving a formula for pattern densities in tensor products and provides explicit examples, including the shortest ones of length 17.

Contribution

It introduces a general formula for pattern densities in permutation tensor products and fully characterizes 3-inflatable permutations, including explicit examples.

Findings

Derived a formula for limit pattern densities in permutation tensor products.

Complete characterization of 3-inflatable permutations.

Constructed explicit examples of 3-inflatable permutations, shortest being length 17.

Abstract

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 $k \geq 4$. 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 $17$.