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A recursive method for the oddness of the number of set-valued tableaux

Taikei Fujii, Takahiko Nobukawa, Tatsushi Shimazaki

TL;DR

The paper addresses the parity of $|{ m SVT}( heta,n)|$ for skew diagrams, showing it is always odd. It develops a recursive, constructive induction by removing the bottom-right box and classifying extensions, complemented by a sign-reversing involution to cancel even contributions. The result highlights a robust oddness property for set-valued tableaux and connects to the tableau-sum framework for Grothendieck polynomials, with multiple alternative proofs provided in appendices. These findings enrich the combinatorial understanding underpinning K-theoretic Schubert calculus and offer versatile methodological perspectives for parity arguments in tableau combinatorics.

Abstract

Set-valued tableaux, introduced by Buch to express the tableaux-sum formula for stable Grothendieck polynomials, generalize semistandard tableaux. We provide a new recursive proof that the number of set-valued tableaux of a given shape is odd.

A recursive method for the oddness of the number of set-valued tableaux

TL;DR

The paper addresses the parity of for skew diagrams, showing it is always odd. It develops a recursive, constructive induction by removing the bottom-right box and classifying extensions, complemented by a sign-reversing involution to cancel even contributions. The result highlights a robust oddness property for set-valued tableaux and connects to the tableau-sum framework for Grothendieck polynomials, with multiple alternative proofs provided in appendices. These findings enrich the combinatorial understanding underpinning K-theoretic Schubert calculus and offer versatile methodological perspectives for parity arguments in tableau combinatorics.

Abstract

Set-valued tableaux, introduced by Buch to express the tableaux-sum formula for stable Grothendieck polynomials, generalize semistandard tableaux. We provide a new recursive proof that the number of set-valued tableaux of a given shape is odd.
Paper Structure (11 sections, 9 theorems, 84 equations)

This paper contains 11 sections, 9 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions and Young diagrams
  4. Semistandard tableaux
  5. Set-valued tableaux
  6. Skew Young diagrams and set-valued skew tableaux
  7. The odd property for $|\mathrm{SVT}(\theta,n)|$
  8. Conclusions
  9. Proof by using the bi-alternant formula
  10. A sign-reversing involution
  11. A simple proof for the oddness

Key Result

Lemma 3.1

Let $\theta$ be a skew Young diagram. The cardinality of the set of set-valued tableaux of shape $\theta$ satisfies the following: Furthermore, for any $T' \in \mathrm{SVT}(\theta',n)$, its cardinality $|U(T')|$ is odd if the set $U(T')$ is non-empty.

Theorems & Definitions (33)

  • Definition 2.1: [Buc02]
  • Example 2.1
  • Example 2.2
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 23 more