Families of major index distributions: closed forms and unimodality

2018/08/03 by Keith, William J.
#05A05 #05A15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.01362

Abstract

Closed forms for f_λ,i (q) := ∑_{τ∈ SYT(λ) : des(τ) = i} q^maj(τ), 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 λ 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_λ(1,q,q²,…,q^n-1) has a combinatorial realization as the distribution of the major index over a given set of tableaux.

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