A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape

TL;DR

This paper explores special classes of tableau sequences ending with a row shape, revealing connections to Baxter numbers and proposing conjectures linking oscillating and Young tableaux, supported by generating function analysis.

Contribution

It introduces new Baxter classes of tableau sequences ending with a row shape and conjectures a bijection with Young tableaux of bounded height, supported by partial proofs.

Findings

Hesitating tableaux of height ≤ 2 ending with a row are counted by Baxter numbers.

Three new Baxter classes are defined without obvious antipodal symmetry.

Conjecture that oscillating tableaux of height ≤ k ending in a row are bijective with Young tableaux of height 2k, proven for k ≤ 8.

Abstract

Tableau sequences of bounded height have been central to the analysis of k-noncrossing set partitions and matchings. We show here that familes of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. We then conjecture that oscillating tableau of height bounded by k ending in a row are in bijection with Young tableaux of bounded height 2k. We prove this conjecture for k at most eight by a generating function analysis. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found.