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The number of monotone trapezoids with prescribed bottom row

Ilse Fischer, Hans Höngesberg

TL;DR

The paper derives a unified operator framework for counting (h,n)-monotone trapezoids with prescribed bottom rows, generalizing the known theory for monotone triangles and Gelfand–Tsetlin patterns. The central result expresses MT_h(\mathbf{k}_n) as an operator-augmented GT count: MT_h(\mathbf{k}_n) = ∏_{1≤i<j≤n} Strict^{-1}_{k_i,k_j} A_{k_i,k_j} GT_h(\mathbf{k}_n), where A(x,y) is a carefully constructed hidden formal power series that annihilates certain symmetric components and interacts with a Strict operator to cancel non-contributing terms. A Pfaffian formula for GT_h(\mathbf{k}_n) is proved, and a detailed, multi-layered proof shows how telescoping, symmetric/antisymmetric decompositions, and a new annihilating operator lead to the main result; a free top boundary (unprescribed top row) is a novel feature. The work also illuminates a connection to the coinvariant algebra and proposes a conjectural annihilator ideal for the generalized GT_h, suggesting avenues for Littlewood-type identities and deeper algebraic structure in these combinatorial objects.

Abstract

We establish an operator formula for the number of monotone trapezoids with prescribed bottom row, generalizing alternating sign matrices. The special case of the formula for monotone triangles previously provided an alternative proof for the enumeration of alternating sign matrices and led to several results on alternating sign triangles and alternating sign trapezoids. The generalization presented in this paper reveals an additional ``hidden operator'' that is annihilated in the special case of monotone triangles, whose discovery was a major challenge. The enumeration formula is conceptually simple: it applies, in addition to the newly discovered hidden operator, an operator ensuring row strictness to the formula for the number of Gelfand--Tsetlin trapezoids. Notably, the top row of the monotone trapezoid is not prescribed. Thus, our result involves a free boundary, which is a novel situation in this area. We also uncover an unexpected relation to the coinvariant algebra and propose a conjecture on its generalization.

The number of monotone trapezoids with prescribed bottom row

TL;DR

The paper derives a unified operator framework for counting (h,n)-monotone trapezoids with prescribed bottom rows, generalizing the known theory for monotone triangles and Gelfand–Tsetlin patterns. The central result expresses MT_h(\mathbf{k}_n) as an operator-augmented GT count: MT_h(\mathbf{k}_n) = ∏_{1≤i<j≤n} Strict^{-1}_{k_i,k_j} A_{k_i,k_j} GT_h(\mathbf{k}_n), where A(x,y) is a carefully constructed hidden formal power series that annihilates certain symmetric components and interacts with a Strict operator to cancel non-contributing terms. A Pfaffian formula for GT_h(\mathbf{k}_n) is proved, and a detailed, multi-layered proof shows how telescoping, symmetric/antisymmetric decompositions, and a new annihilating operator lead to the main result; a free top boundary (unprescribed top row) is a novel feature. The work also illuminates a connection to the coinvariant algebra and proposes a conjectural annihilator ideal for the generalized GT_h, suggesting avenues for Littlewood-type identities and deeper algebraic structure in these combinatorial objects.

Abstract

We establish an operator formula for the number of monotone trapezoids with prescribed bottom row, generalizing alternating sign matrices. The special case of the formula for monotone triangles previously provided an alternative proof for the enumeration of alternating sign matrices and led to several results on alternating sign triangles and alternating sign trapezoids. The generalization presented in this paper reveals an additional ``hidden operator'' that is annihilated in the special case of monotone triangles, whose discovery was a major challenge. The enumeration formula is conceptually simple: it applies, in addition to the newly discovered hidden operator, an operator ensuring row strictness to the formula for the number of Gelfand--Tsetlin trapezoids. Notably, the top row of the monotone trapezoid is not prescribed. Thus, our result involves a free boundary, which is a novel situation in this area. We also uncover an unexpected relation to the coinvariant algebra and propose a conjecture on its generalization.

Paper Structure

This paper contains 13 sections, 21 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of Theorem \ref{['GT']}
  4. Sketch of the proof of Theorem \ref{['main']}
  5. The hidden formal power series
  6. First part of the proof of Theorem \ref{['main']}
  7. Proofs of Lemmas \ref{['one']} and \ref{['zero']}
  8. Proof of Lemma \ref{['one']}
  9. Towards proving Lemma \ref{['zero']}
  10. An additive decomposition
  11. Proof of Lemma \ref{['zero']}
  12. The case n+h being odd
  13. A conjecture on the annihilator ideal

Key Result

Theorem 4

Let $h$ and $n$ be non-negative integers with $n \ge h$.

Theorems & Definitions (45)

  • Definition 1
  • Example 2
  • Example 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Example 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 35 more