Major index over descent for pattern-avoiding permutations

William J. Keith

Published 2017-07-05Version 1

An open conjecture in pattern avoidance theory is that the distribution of the major index among 321-avoiding permutations is distributed unimodally. We construct a formula for this distribution, and in the case of 2 descents prove unimodality, with unimodality for 3 through 5 descents likely being little more complicated. The formula refines the $q$-analogue of the Frame-Robinson-Thrall hooklength formula for two-rowed partitions, and in the latter part of the paper we discuss further conjecture and exploration toward this question. We also give observations on the analogous behaviors for other permutation patterns of length 3.