[Paper Review] Alternating sign trapezoids and a constant term approach

Ilse Fischer|arXiv (Cornell University)|Apr 23, 2018

Advanced Combinatorial Mathematics18 references3 citations

TL;DR

This paper proves a conjecture by Behrend and Aigner that the number of $(n,l)$-alternating sign trapezoids equals the number of column strict shifted plane partitions of class $l-1$ with at most $n$ parts in the top row, using a constant term approach based on operator formulas for monotone triangles. It further establishes that three statistics—$\operatorname{r}(T)$, $\operatorname{p}(T)$, and $\operatorname{q}(T)$—have the same joint distribution on both families, suggesting a potential bijection.

ABSTRACT

We show that there is the same number of (n,l)-alternating sign trapezoids as there is of column strict shifted plane partitions of class l-1 with at most n parts in the top row, thereby proving a result that was conjectured independently by Behrend and Aigner. The first objects generalize alternating sign triangles, which have recently been introduced by Ayyer, Behrend and the author who showed that they are counted by the same formula as alternating sign matrices. Column strict shifted plane partitions of a fixed class were introduced in a slightly different form by Andrews, and they essentially generalize descending plane partitions. They also correspond to cyclically symmetric lozenge tilings of a hexagon with a triangular hole in the center. In addition, we also provide three statistics on each class of objects and show that their joint distribution is the same. We prove our result by employing a constant term approach that is based on the author's operator formula for monotone triangles. This paper complements a forthcoming paper of Behrend and the author, where the six-vertex model approach is used to show equinumeracy as well as generalizations involving statistics that are different from those considered in the present paper.

Motivation & Objective

• To prove the equinumerosity conjecture by Behrend and Aigner relating $(n,l)$-alternating sign trapezoids and column strict shifted plane partitions of class $l-1$ with at most $n$ parts in the top row.
• To establish a joint distribution equivalence of three statistics—$\operatorname{r}(T)$, $\operatorname{p}(T)$, $\operatorname{q}(T)$—on both classes of objects, hinting at a deeper structural connection.
• To employ a constant term approach rooted in operator formulas for monotone triangles to derive generating functions for both combinatorial families.
• To complement a forthcoming six-vertex model-based proof by Behrend and the author by providing an alternative, non-bijective route to equinumerosity with different statistics.

Proposed method

• The constant term approach is applied to a matrix determinant derived from an operator formula for monotone triangles, transforming the enumeration problem into a generating function computation.
• The determinant is manipulated using algebraic identities, including the Chu-Vandermonde summation and binomial coefficient duality via $\binom{n}{k} = (-1)^k \binom{k-n-1}{k}$.
• The matrix entries are re-expressed in terms of lattice path generating functions, where weights correspond to path features such as horizontal steps at height 0 and diagonal crossings.
• The Lindström-Gessel-Viennot theorem is used to interpret the determinant as a generating function for non-intersecting lattice paths, which are then bijectively linked to column strict shifted plane partitions.
• The weight function incorporates $P$, $Q$, and $R$ parameters to encode statistics on $10$-columns and $1$-columns in alternating sign trapezoids, with special treatment for the central column when $d=0$.
• For $d \geq 1$, the weight encodes whether the diagonal $y = x + d$ is crossed via a north or west step; for $d = 0$, the weight includes a ternary term $(P+Q-1)$ for paths crossing the origin on the diagonal.

Experimental results

Research questions

• RQ1 Are the numbers of $(n,l)$-alternating sign trapezoids and column strict shifted plane partitions of class $l-1$ with at most $n$ parts in the top row equal?
• RQ2 Do the three statistics $\operatorname{r}(T)$, $\operatorname{p}(T)$, and $\operatorname{q}(T)$ on alternating sign trapezoids have the same joint distribution as their counterparts on column strict shifted plane partitions?
• RQ3 Can the constant term method based on operator formulas for monotone triangles be used to derive generating functions that enumerate both classes of objects?
• RQ4 How do the weights in the lattice path interpretation encode the statistics on alternating sign trapezoids, especially in the case $d=0$?

Key findings

• The number of $(n,l)$-alternating sign trapezoids is equal to the number of column strict shifted plane partitions of class $l-1$ with at most $n$ parts in the top row, confirming the conjecture by Behrend and Aigner.
• The joint distribution of the statistics $\operatorname{r}(T)$, $\operatorname{p}(T)$, and $\operatorname{q}(T)$ is identical on both classes of objects, suggesting a possible bijection.
• The generating function for alternating sign trapezoids is expressed as a determinant involving binomial coefficients weighted by $P$, $Q$, and $R$, which matches the generating function for column strict shifted plane partitions.
• For $P=Q=R=1$, the determinant reduces to a known form that enumerates column strict shifted plane partitions of class $l-1$, confirming consistency with established results.
• The lattice path interpretation via the Lindström-Gessel-Viennot theorem provides a combinatorial model where path features correspond directly to the statistics on trapezoids, especially for $d \geq 1$.
• In the case $d=0$, the weight function includes a ternary factor $(P+Q-1)$ for paths crossing the main diagonal at the origin, which correctly encodes the central column's contribution to the statistics.