Symmetry of ascent and descent distributions on rectangular and staircase tableaux

TL;DR

This paper provides direct bijective proofs of the symmetry between ascent and descent distributions in standard Young tableaux of rectangular and staircase shapes, connecting to various combinatorial structures and conjectures.

Contribution

It introduces new bijections based on arrow encodings that prove symmetry, relate to known involutions, and address conjectures in tableau and permutation statistics.

Findings

Proves symmetry of ascent and descent distributions in specific tableaux shapes.

Establishes connections with Dyck path involutions and Narayana numbers.

Provides bijective proofs for conjectures on permutation statistics.

Abstract

We give direct bijective proofs of the symmetry of the distributions of the number of ascents and descents over standard Young tableaux of shape $\lambda$, where $\lambda$ is a rectangle $(n,n,\dots,n)$ or a truncated staircase $(n,n-1,\dots,n-k+1)$. These can be viewed as instances of the more general symmetry of the distribution of descents over linear extensions of graded posets, for which previous proofs by Stanley and Farley were based on the theory of $P$-partitions and the involution principle, respectively. In the case of two-row rectangles $(n,n)$, our bijection is equivalent to the Lalanne--Kreweras involution on Dyck paths, which bijectively proves the symmetry of the Narayana numbers. Our bijections are defined in terms of certain arrow encodings of standard Young tableaux. This setup allows us to construct other statistic-preserving involutions on tableaux of rectangular shape, providing a simple proof of the fact that ascents and descents are equidistributed up to a shift, and proving a conjecture of Sulanke about certain statistics in the case of three rows. Finally, we use our bijections to define a possible notion of rowmotion on standard Young tableaux of rectangular shape, and to give a bijective proof of the symmetry of the number of descents on canon permutations, which have been recently studied as a variation of Stirling and quasi-Stirling permutations.