Families of major index distributions: closed forms and unimodality

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.

20 pages. Portions of material presented at Joint Meetings 2018 and at Combinatory Analysis 2018 (GEA80)