3 problems

Let $\mathfrak M^{(n)}(q)$ and $\widetilde{\mathfrak M}^{(n)}(q)$ be the matrices whose entries are respectively the sums of $q^{{\rm tot}(\sigma)}$ over permutations with…

Let $\mathcal T_n$ denote the set of tree-like tableaux of size $n$, and let a corner be a corner of such a tableau. Corner enumeration conjecture. The total number of corners amon…

Consider the two-species PASEP at $q=1$. Let $X$ be a state represented by a word in $\{\operatorname{D},\operatorname{E},\operatorname{A}\}^n$ with precisely $r$…