Enumeration of standard barely set-valued tableaux of shifted shapes
Jang Soo Kim, Michael J. Schlosser, Meesue Yoo

TL;DR
This paper derives a formula for counting standard barely set-valued tableaux of shifted shapes using $q$-integral techniques, and applies it to confirm conjectures related to their enumeration and properties in Young's lattice.
Contribution
It provides a new formula for counting these tableaux of arbitrary shifted shapes and verifies conjectures on their enumeration and down-degree expectations.
Findings
Derived a general counting formula for shifted barely set-valued tableaux.
Confirmed conjectures on the enumeration of specific shifted-balanced shapes.
Proved a conjecture on the CDE property of a particular shifted shape.
Abstract
A standard barely set-valued tableau of shape is a filling of the Young diagram with integers such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to the coincidental down-degree expectations (CDE) of lower intervals in Young's lattice. Using -integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shape. We show how it can be used to recover two formulas, originally conjectured by Reiner, Tenner and Yong, and proved by Hopkins, for numbers of standard barely set valued tableaux of particular shifted-balanced shapes. We also prove a conjecture of Reiner, Tenner and Yong on the CDE property of the shifted shape…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Topological and Geometric Data Analysis
