5 problems

For integers $\ell,r\geqslant1$, let $\gamma^2(\ell,r)$ denote the asymptotic variance associated with the tree permutation determined by the parameters $\ell$ and $r$. The compute…

Let $\boldsymbol{\tau}_n$ be a uniformly random tree permutation of length $n$, and let $\boldsymbol{\tau}_n$ have a fixed forest pattern $\sigma$ with $d\geqslant2$ blocks, each o…

For a pattern $\xi\in S_k$, let $c_n^\xi(x)$ denote the pattern-occurrence generating polynomial over $S_n$, so that its coefficients enumerate permutations according to the number…

Fix $n,k\in\mathbb N$ and a pattern $v\in[k]^*$ using $d>1$ distinct letters. Let $M_r$ be the maximal number of states with $d-1$ loops in a simple path in the automaton…

vanishing-coefficient conjecture. There exist $k,r\in\mathbb N$ and a pattern $v\in[k]^*$ such that $d>1$ and $K_r=0$. In that case, there exist $L_r\in\mathbb N$, $L_r<M_r$, and…