6 problems

Let $B$ be a board, let $G(B)$ be its rook equivalence graph, and suppose that $\{B_1,B_2\}$ is an edge of $G(B)$. For a nonnegative integer $k$, consider the placements of $k$ roo…

Let $K_{a,b}$ be a complete bipartite graph, and let $B$ be a Ferrers board whose rook equivalence graph is $K_{a,b}$. Complete bipartite rook equivalence conjecture. Then $a=b=1$.…

Let $w\in\mathfrak{S}_n$ be a $123$-avoiding permutation, meaning that it contains no subsequence order-isomorphic to $123$, and let $I_w$ be its diagram. Write…

For a positive integer $n$, let $v=(2n-1)(2n)(2n-3)(2n-2)\cdots3412$ be the indicated permutation, and let $I_v$ be its diagram. Write $M_{2n}(\overline{I_v},q)$ for the matrix cou…

Let $B\subseteq[m]\times[n]$ be a board, let $r$ be a rank, and let $q$ be a prime power. The $q$-hit number $H_r(B,q)$ is defined for this board and rank. Weak positivity conjectu…

Let $Y(n,s)$ be the set of all Young diagrams with $n$ rows of lengths between $s$ and $n+s$. For a rook placement $C$ on a diagram $[?]$ in $Y(n,s)$, let $\operatorname{inv}(C)$ b…