The major index (maj) and its Sch\"utzenberger dual

TL;DR

This paper constructs a bijective proof for the generating function of semistandard Young tableaux of skew shape, linking it to a particle system representation and revealing a relation between tableau statistics via Schützenberger involution.

Contribution

It introduces a novel bijection involving plinths and reading Young diagrams, providing new insights into the combinatorics of skew shape tableaux and their generating functions.

Findings

Bijective proof of Stanley's formula for skew shape SsYT

Introduction of the plinth and its properties

Relation between volume and major index via Schützenberger involution

Abstract

We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $\lambda/\mu.$ The partition function of this particle system gives the generating function of the SsYT of skew shape $\lambda/\mu.$ Thus we obtain a bijective proof of the Stanley formula for the SsYT generating function. To do this we define for every SsYT $T$ its plinth, $\mathsf{p}\left( T\right) ,$ which is a SsYT of the same shape $\lambda/\mu.$ The set of plinths is finite. Our bijection associates to every SsYT $T$ a pair $\left( \mathsf{p}\left( T\right) ,Y\left( T-\mathsf{p}\left( T\right) \right) \right) ,$ where $Y\left( T-\mathsf{p}\left( T\right) \right) $ is the reading Young diagram of the SsYT $\left( T-\mathsf{p}\left( T\right) \right) $. \newline In particular, every Standard Young Tableau (SYT) $P$ has its plinth, $\mathsf{p}\left( P\right) $. The two statistics of SYT-s -- the volume $\left\vert \mathsf{p}\left( P\right) \right\vert $ and $\mathsf{maj}\left( P\right) $ -- are related via the Sch\"{u}tzenberger involution $Sch:$% \[ \left\vert \mathsf{p}\left( P\right) \right\vert =\mathsf{maj}\left( Sch\left( P\right) \right) . \]