Families of Major Index Distributions: Closed Forms and Unimodality

William J Keith

doi:10.37236/8585

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.

Highlights

In this paper we establish formulas for the distribution of the major index of standard Young tableaux of fixed partition shape and number of descents, for numerous classes of each
This work was initially motivated by a conjecture of Bruce Sagan and collaborators on the unimodality of the distribution of the major index over 321-avoiding permutations of length n with fixed numbers of descents, which we prove : Conjecture 1. (Sagan et al, [6]) In the generating function n−1 qmaj(σ)tdes(σ) =
Let f(∗n,k)\(j),i denote the distribution of the major index over skew standard Young tableaux of shape (n, k) \ (j) with i descents in which the entry 1 is in the top row

Summary

Introduction

In this paper we establish formulas for the distribution of the major index of standard Young tableaux of fixed partition shape and number of descents, for numerous classes of each. Kirillov and Reshetikhin’s result shows that the generating function of the major index of tableaux of a fixed shape and number of descents is a unimodal polynomial. Polynomiality and positivity of the formulas described, which are not always obviously nonnegative or polynomials, are given by the establishment of this combinatorial description, and unimodality by invoking the result of Kirillov and Reshetikhin In many of these cases, we can establish unimodality by a second means of independent interest, identifying these distributions as a power of q times the principal specializations of particular Schur polynomials. The distribution of the major index over all skew two-rowed standard Young tableaux of a given shape with fixed number of descents is a unimodal polynomial.

Definitions and Background

The Kirillov-Reshetikhin theorem

Formulas

Three-rowed tableaux

Future work