3 problems

Let $4\pi\in\mathfrak{S}_{64}4$ be a permutation that is 4-inflatable, meaning that it has the equal-pattern-count property required for the paper's 4-inflatable permutations. Say…

Let $4\mathfrak{S}_n4$ be the symmetric group on $n$ symbols. For $4\tau\in\mathfrak{S}_k4$ and $4\sigma\in\mathfrak{S}_n4$, let $4\mathbf{X}^{\tau}(\sigma)4$ count the occurrences…

For a permutation, a pattern being uniformly distributed on $k$ symbols need not imply uniform distribution on $(k+1)$ symbols. The upward-implication conjecture. This failure, kno…