Bijecting hidden symmetries for skew staircase shapes

TL;DR

This paper introduces a new bijection linking specific Young tableaux shapes, providing combinatorial proofs for formulas, symmetry properties, and enabling efficient sampling, with applications in K-theory and related areas.

Contribution

It presents a novel bijection between staircase minus rectangle and shifted Young tableaux shapes, resolving an open problem and extending to set-valued tableaux in K-theory.

Findings

Provides a bijective proof of the product formula for staircase minus rectangle tableaux

Establishes a bijection for semistandard Young tableaux and symmetry of LR-coefficients

Extends results to set-valued tableaux in K-theory, offering new proofs

Abstract

We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for the number of standard Young tableaux of staircase minus rectangle shape. This resolves an open problem by Morales, Pak and Panova (2019), and allows for efficient random sampling. Other applications include a bijection for semistandard Young tableaux, and a bijective proof of Stembridge's symmetry of LR-coefficients of the staircase shape. We also extend these results to set-valued standard Young tableaux in the combinatorics of K-theory, leading to new proofs of results by Lewis and Marberg (2019) and Abney-McPeek, An and Ng (2020).