Additional refined-statistic involutions for three-row tableaux

Construct involutions \(\alpha,\delta:\SYT(n^3)\to\SYT(n^3)\) such that \(\alpha\) swaps \(b_{1,1}\) with \(b_{2,1}\), \(b_{1,2}\) with \(b_{2,3}\), \(b_{1,3}\) with \(b_{2,2}\), and \(b_{3,2}\) with \(b_{3,3}\) while preserving \(b_{3,1}\), and \(\delta\) swaps \(b_{2,1}\) with \(b_{2,3}-1\) while preserving \(b_{1,2}\), \(b_{2,2}\), and \(b_{3,2}\).

Background

The paper reports computational evidence suggesting that the joint distribution of the refined bounce statistics $b_{r,s}$ on $\SYT(n^3)$ has additional symmetries beyond those proved by the involutions constructed earlier.

The conjectured involutions $\alpha$ and $\delta$ would each imply the unresolved $\st_1$ part of Sulanke’s conjecture. Their existence is therefore a stronger structural problem concerning refined descent statistics on three-row rectangular tableaux.