On the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e268" altimg="si6.svg"><mml:mi>q</mml:mi></mml:math>-enumeration of barely set-valued tableaux and plane partitions

Barely set-valued tableaux are a variant of Young in which one box contains two numbers as its entry. It has recently been discovered that there product formulas enumerating certain classes barely tableaux. We give some q-analogs these by introducing version major index for also and plane partitions. Many the results stated generality P-partitions then specialize to particularly nice rectangles minuscule posets. The proofs use several probability distributions on set order ideals poset, depending real parameter q>0, we think could be independent interest.