Families of major index distributions: closed forms and unimodality

TL;DR

This paper derives closed-form formulas for the distribution of the major index over standard Young tableaux with fixed shape and descents, establishing unimodality and connections to Schur functions.

Contribution

It provides new closed-form expressions for major index distributions for various shapes, confirming unimodality and linking to Schur functions via Jacobi-Trudi identities.

Findings

Formulas are unimodal for large classes of shapes.

Many formulas are specializations of Schur functions.

Finite principal Schur function specializations have combinatorial interpretations.

Abstract

Closed forms for $f_{\lambda,i} (q) := \sum_{\tau \in SYT(\lambda) : des(\tau) = i} q^{maj(\tau)}$, the distribution of the major index over standard Young tableaux of given shapes and specified number of descents, are established for a large collection of $\lambda$ and $i$. Of particular interest is the family that gives a positive answer to a question of Sagan and collaborators. All formulas established in the paper are unimodal, most by a result of Kirillov and Reshetikhin. Many can be identified as specializations of Schur functions via the Jacobi-Trudi identities. If the number of arguments is sufficiently large, it is shown that any finite principal specialization of any Schur function $s_\lambda(1,q,q^2,\dots,q^{n-1})$ has a combinatorial realization as the distribution of the major index over a given set of tableaux.