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Staircase hook-length ratios and special values of Jacobi polynomials

Tatsushi Shimazaki

TL;DR

This work ties the combinatorics of staircase hook-length ratios to classical special functions by showing that the ratio of semistandard tableau counts for adjacent staircase shapes equals a Jacobi polynomial value at $-1$, with parameters tied to $n$ and $k$. This hypergeometric viewpoint yields explicit recurrences for stable Grothendieck polynomials and $K$-theoretic Schur $P$-functions via a Holman-type hypergeometric function, and it is illuminated by a unified excited Young diagram framework. The results bridge tableaux counting, symmetric functions in $K$-theory, and orthogonal polynomials, and reveal a consistent combinatorial interpretation through excitations in both ordinary and shifted settings, including a notable $3$-power factor in shifted shapes. The approach provides new structural insight into the staircase case and suggests $q$-deformations and broader connections within symmetric function theory.

Abstract

We relate hook-length products for adjacent staircase partitions to special values of Jacobi polynomials. This connection expresses the number of semistandard tableaux in terms of Jacobi polynomials defined via Gauss hypergeometric functions. From this identity, we derive the special values of stable Grothendieck polynomials and $K$-theoretic Schur $P$-functions indexed by adjacent staircase partitions. These values provide ratios of the numbers of set-valued and shifted set-valued semistandard tableaux. This connection is further clarified by the theory of excited Young diagrams, which characterizes the coefficients in these specializations.

Staircase hook-length ratios and special values of Jacobi polynomials

TL;DR

This work ties the combinatorics of staircase hook-length ratios to classical special functions by showing that the ratio of semistandard tableau counts for adjacent staircase shapes equals a Jacobi polynomial value at , with parameters tied to and . This hypergeometric viewpoint yields explicit recurrences for stable Grothendieck polynomials and -theoretic Schur -functions via a Holman-type hypergeometric function, and it is illuminated by a unified excited Young diagram framework. The results bridge tableaux counting, symmetric functions in -theory, and orthogonal polynomials, and reveal a consistent combinatorial interpretation through excitations in both ordinary and shifted settings, including a notable -power factor in shifted shapes. The approach provides new structural insight into the staircase case and suggests -deformations and broader connections within symmetric function theory.

Abstract

We relate hook-length products for adjacent staircase partitions to special values of Jacobi polynomials. This connection expresses the number of semistandard tableaux in terms of Jacobi polynomials defined via Gauss hypergeometric functions. From this identity, we derive the special values of stable Grothendieck polynomials and -theoretic Schur -functions indexed by adjacent staircase partitions. These values provide ratios of the numbers of set-valued and shifted set-valued semistandard tableaux. This connection is further clarified by the theory of excited Young diagrams, which characterizes the coefficients in these specializations.
Paper Structure (13 sections, 11 theorems, 100 equations, 1 table)

This paper contains 13 sections, 11 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions, Young diagrams, and hook lengths
  4. Gauss hypergeometric function
  5. Jacobi polynomials
  6. A hypergeometric identity for semistandard tableaux
  7. Applications to Grothendieck polynomials and $K$-theoretic Schur $P$-functions
  8. Jacobi polynomial recurrences for Grothendieck polynomials via set-valued tableaux
  9. Jacobi polynomial recurrences for $K$-theoretic Schur $P$-functions via shifted set-valued tableaux
  10. A combinatorial interpretation via excited Young diagrams
  11. Representation of Schur and Grothendieck polynomials by excited Young diagrams
  12. Representation of $K$-theoretic Schur $P$-functions by excited Young diagrams
  13. Conclusion

Key Result

Lemma 3.1

Let $\delta_k=(k-1,k-2,\dots,1)$ be the staircase partition. For positive integers $k$ and $n$ with $n\ge k$, the following hold.

Theorems & Definitions (34)

  • Example 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Example 3.1
  • ...and 24 more