On the $q$-Enumeration of Barely Set-Valued Tableaux and Plane Partitions

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 q-analogs of these product formulas by introducing a version of major index for these tableaux. We also give product formulas and q-analogs for barely set-valued plane partitions. The proofs use several probability distributions on the set of order ideals of a poset, depending on the real parameter q > 0, which we think could be of independent interest.

Homomesy via Toggleability Statistics

“…As a result, showing f ≡ const for certain f can lead to enumerative results about tableaux: see [28]. Moreover, showing f ≡ q const(q) can similarly yield results about the q-enumeration of tableaux: see the very recent paper [19].…”