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Yet another doubly refined enumeration of Alternating Sign Matrices

Guo-Niu Han, Lihong Yang

TL;DR

The work develops an explicit determinantal generating function for the doubly refined enumeration of alternating sign matrices with respect to the number of $-1$’s and the positions of the boundary $1$’s, using Izergin–Korepin theory for the six-vertex model together with Lascoux’s symmetry-function determinant. The main result expresses $A_n(z,\rho,\tau)$ as a determinant $\det\bigl(K^{\rho,\tau}_{i,j}(q)\bigr)$ up to a prefactor, after a careful specialization and determinant manipulation, providing a new, compact determinantal formula that unifies and extends prior refinements. The authors derive several corollaries and compare them with known results (e.g., Aigner, BEZ), showing consistency and offering new identities. They also refine the decomposition conjecture of MRR/Kuperberg by introducing a $\rho$-dependent factorization, supported by computed cases up to $n\le 9$. The approach opens a cohesive framework linking the six-vertex model, symmetric-function determinantal identities, and refined ASM enumeration.

Abstract

Since the alternating sign matrix conjecture, proposed by Mills, Robbins, and Rumsey in 1982, was proved by Zeilberger and Kuperberg, several refined enumerations have been considered. In particular, Behrend et al. obtained a quadruply refined enumeration by adding certain parameters. In this paper, we revisit the doubly refined enumeration of alternating sign matrices by adding three parameters: the number of $-1$'s, the position of the $1$ in the first row, and the position of the $1$ in the last row. Using Lascoux's formula on symmetry functions, we derive a new determinantal formula for this doubly refined enumeration. Besides the enumeration conjecture, Mills et al. also proposed a decomposition conjecture, which was subsequently proven by Kuperberg. We present a refinement of that decomposition conjecture.

Yet another doubly refined enumeration of Alternating Sign Matrices

TL;DR

The work develops an explicit determinantal generating function for the doubly refined enumeration of alternating sign matrices with respect to the number of ’s and the positions of the boundary ’s, using Izergin–Korepin theory for the six-vertex model together with Lascoux’s symmetry-function determinant. The main result expresses as a determinant up to a prefactor, after a careful specialization and determinant manipulation, providing a new, compact determinantal formula that unifies and extends prior refinements. The authors derive several corollaries and compare them with known results (e.g., Aigner, BEZ), showing consistency and offering new identities. They also refine the decomposition conjecture of MRR/Kuperberg by introducing a -dependent factorization, supported by computed cases up to . The approach opens a cohesive framework linking the six-vertex model, symmetric-function determinantal identities, and refined ASM enumeration.

Abstract

Since the alternating sign matrix conjecture, proposed by Mills, Robbins, and Rumsey in 1982, was proved by Zeilberger and Kuperberg, several refined enumerations have been considered. In particular, Behrend et al. obtained a quadruply refined enumeration by adding certain parameters. In this paper, we revisit the doubly refined enumeration of alternating sign matrices by adding three parameters: the number of 's, the position of the in the first row, and the position of the in the last row. Using Lascoux's formula on symmetry functions, we derive a new determinantal formula for this doubly refined enumeration. Besides the enumeration conjecture, Mills et al. also proposed a decomposition conjecture, which was subsequently proven by Kuperberg. We present a refinement of that decomposition conjecture.
Paper Structure (7 sections, 18 theorems, 95 equations, 2 figures)

This paper contains 7 sections, 18 theorems, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Six-vertex model
  3. Lascoux's formula
  4. Proof of Theorem \ref{['th:main']}
  5. Applications and Corollaries
  6. Comparison with known results
  7. A decomposition conjecture

Key Result

Theorem 1.1

The generating function for the $(2+q+q^{-1})$-enumeration of doubly-refined ASMs is where

Figures (2)

  • Figure 1: The six vertex states, their corresponding matrix entries, and associated weights.
  • Figure 2: The square ice configuration corresponding to the ASM in (\ref{['m:asm1']}).

Theorems & Definitions (24)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1: Izergin, Korepin
  • Theorem 3.1: Lascoux
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 14 more