Families of major index distributions: closed forms and unimodality

William J. Keith¹

¹Michigan Technological University

The electronic journal of combinatorics, Tome 26 (2019) no. 3

Résumé

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.

Détail

DOI : 10.37236/8585

Classification : 05E10, 05A17, 11P83
Mots-clés : quantum inverse scattering, Bethe's ansatz, integrable quantum systems, standard Young tableaux, rigged configurations