Generalized Monotone Triangles: an extended Combinatorial Reciprocity Theorem

TL;DR

This paper introduces Generalized Monotone Triangles, extending previous combinatorial objects, and proves their connection to polynomial evaluations, offering new insights and potential bijective proofs related to Alternating Sign Matrices.

Contribution

It defines Generalized Monotone Triangles as a joint extension of Monotone and Decreasing Monotone Triangles and proves their enumeration corresponds to polynomial evaluations.

Findings

Evaluation of alpha(n;k1,...,kn) counts Generalized Monotone Triangles.

Certain polynomial evaluations relate to well-known combinatorial numbers.

Main theorem provides a new combinatorial interpretation and potential proof strategies.

Abstract

In a recent work, the combinatorial interpretation of the polynomial alpha(n;k1,k2,...,kn) counting the number of Monotone Triangles with bottom row k1 < k2 < ... < kn was extended to weakly decreasing sequences k1 >= k2 >= ... >= kn. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles - a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of alpha(n;k1,k2,...,kn) at arbitrary (k1,k2,...,kn) in Z^n is a signed enumeration of Generalized Monotone Triangles with bottom row (k1,k2,...,kn). Computational experiments indicate that certain evaluations of the polynomial at integral sequences yield well-known round numbers related to Alternating Sign Matrices. The main result provides a combinatorial interpretation of the conjectured identities and could turn out useful in giving a bijective proof.