An Extension of the Foata Map to Standard Young Tableaux

TL;DR

This paper introduces an inversion statistic for standard Young tableaux and demonstrates its equidistribution with the major index by constructing a bijection inspired by the Foata map, extending combinatorial tools to tableaux.

Contribution

It extends the Foata map concept to standard Young tableaux by defining a new inversion statistic and proving its distributional equivalence with the major index.

Findings

Inversion statistic on SYT is equidistributed with the major index.

Constructed a bijection on SYT analogous to the Foata map.

Established a new combinatorial correspondence for tableaux.

Abstract

We define an inversion statistic on standard Young tableaux. We prove that this statistic has the same distribution over SYT(\lambda) as the major index statistic by exhibiting a bijection on SYT(\lambda) in the spirit of the Foata map on permutations.