  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 $17$.
