On the distribution of the major index on standard Young tableaux

TL;DR

This paper investigates the distribution of the major index on standard Young tableaux, introduces partial orders analogous to Bruhat orders, and classifies limit laws and properties like unimodality and log-concavity.

Contribution

It introduces new partial orders on tableaux, classifies major index distributions, and extends limit law results to a broader class of shapes.

Findings

Classified all limit laws for major index on tableaux shapes.

Established partial orders on tableaux analogous to Bruhat orders.

Analyzed unimodality, log-concavity, and local limit properties.

Abstract

The study of permutation and partition statistics is a classical topic in enumerative combinatorics. The major index statistic on permutations was introduced a century ago by Percy MacMahon in his seminal works. In this extended abstract, we study the well-known generalization of the major index to standard Young tableaux. We present several new results. In one direction, we introduce and study two partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. The existence of such ranked poset structures allows us to classify the realizable major index statistics on standard tableaux of arbitrary straight shape and certain skew shapes, and has representation-theoretic consequences, both for the symmetric group and for Shephard-Todd groups. In a different direction, we consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary statistic, "aft," generalizing earlier results of Canfield-Janson-Zeilberger, Chen-Wang-Wang, and others. We also study unimodality, log-concavity, and local limit properties.