Answering Two OPAC Problems Involving Banff Quivers

In a post on the Open Problems in Algebraic Combinatorics (OPAC) blog, E. Bucher and J. Machacek posed three open problems: OPAC-033, OPAC-034, and OPAC-035. These three problems deal with the relationships between three infinite classes of quivers: the Banff, Louise, and $\mathcal{P}$ quivers. OPAC-034 asks whether or not every Banff quiver can be verified to be Banff by only considering sources and sinks, and OPAC-035 asks whether or not every Banff quiver is contained in the class $\mathcal{P}$. We give an answer to both questions, showing that every Banff quiver can be verified to be Banff by using sources and sinks, and therefore that every Banff quiver lives in the class $\mathcal{P}$. We also make some progress on OPAC-033, showing a result similar to our result OPAC-034 for Louise quivers.Comment: 10 pages, 1 figur

Permutations whose reverse shares the same recording tableau in the RSK correspondence

The RSK correspondence is a bijection between permutations and pairs of standard Young tableaux with identical shape, where the tableaux are commonly denoted $P$ (insertion) and $Q$ (recording). It has been an open problem to demonstrate $$|\{w \in \mathfrak{S}_n | \, Q(w) = Q(w^r)\}| = \begin{cases} \displaystyle 2^{\frac{n-1}{2}}{n-1 \choose \frac{n-1}{2}} & n \text{ odd} \newline \displaystyle 0 & n \text{ even} \end{cases},$$ where $w^r$ is the reverse permutation of $w$. First we show that for each $w$ where $Q(w) = Q(w^r)$ the recording tableau $Q(w)$ has a symmetric hook shape and satisfies a certain simple property. From these two results, we succeed in proving the desired identity.Comment: 14 pages, 4 figure