Families of major index distributions: closed forms and unimodality (Q2325759)

Summary: Closed forms for \(f_{\lambda,i} (q) := \sum_{\tau \in \mathrm{SYT}(\lambda):\mathrm{des}(\tau) = i}q^{\mathrm{maj}(\tau)}\), 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 [\textit{A. N. Sergeev}, Math. USSR, Sb. 51, 419--427 (1985; Zbl 0573.17002); translation from Mat. Sb., Nov. Ser. 123(165), No. 3, 422--430 (1984)]. All formulas established in the paper are unimodal, most by a result of \textit{A. N. Kirillov} and \textit{N. Yu. Reshetikhin} [J. Sov. Math. 41, No. 2, 925--955 (1988; Zbl 0639.20029)]. 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.