On the -Enumeration of Barely Set-Valued Tableaux and Plane Partitions
arXiv:2106.07418 · doi:10.1016/j.ejc.2023.103760
Abstract
Barely set-valued tableaux are a variant of Young tableaux in which one box contains two numbers as its entry. It has recently been discovered that there are product formulas enumerating certain classes of barely set-valued tableaux. We give some -analogs of these product formulas by introducing a version of major index for these tableaux. We also give product formulas and -analogs for barely set-valued plane partitions. Many of the results are stated in the generality of -partitions that then specialize to particularly nice formulas for rectangles and minuscule posets. The proofs use several probability distributions on the set of order ideals of a poset, depending on the real parameter , which we think could be of independent interest.
34 pages, 6 tables, 3 figures; v2: Rewrote proof outline in Introduction, rewrote proof of Corollary 2.10, several other minor revisions at the recommendation of referees to improve exposition. To appear in European Journal of Combinatorics