A generalized major index statistic on tableaux

TL;DR

This paper generalizes the major index statistic to standard Young tableaux, connecting existing permutation statistics and answering a question about intermediate statistics.

Contribution

It introduces a family of generalized major index statistics for tableaux, extending previous work and linking permutation and tableau combinatorics.

Findings

Defines a continuum of major index statistics for tableaux

Connects traditional and inversion statistics within this family

Addresses a question posed by Assaf (2008)

Abstract

We extend the family of statistics maj_d, introduced for permutations by Kadell (1985), to standard Young tableaux. At one extreme, we have the traditional Major index statistic maj_1 for tableaux. At the other end, maj_n = inv, the inversion statistic introduced by Haglund and Stevens (2006). This is answers a question of Assaf (2008), who defined maj_2 and maj_3 for tableaux.