Combinatorial Reciprocity for Monotone Triangles

TL;DR

This paper explores the polynomial enumeration of Monotone Triangles, introduces Decreasing Monotone Triangles, and reveals surprising combinatorial identities and connections, including open problems for bijective proofs.

Contribution

It establishes a new combinatorial interpretation for polynomial evaluations of Monotone Triangles and uncovers novel identities linking different classes of these objects.

Findings

Evaluation of the polynomial at decreasing sequences yields signed counts of Decreasing Monotone Triangles.

A key identity relates alpha(n; 1,2,...,n) to alpha(2n; n,...,1).

Decreasing Monotone Triangles correspond to ASM-like matrices, suggesting deep combinatorial connections.

Abstract

The number of Monotone Triangles with bottom row k1 < k2 < ... < kn is given by a polynomial alpha(n; k1,...,kn) in n variables. The evaluation of this polynomial at weakly decreasing sequences k1 >= k2 >= ... >= kn turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects -- in particular it is shown that alpha(n; 1,2,...,n) = alpha(2n; n,n,n-1,n-1,...,1,1). In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing Monotone Triangles with bottom row (n,n,n-1,n-1,...,1,1) is in one-to-one correspondence with a certain set of ASM-like matrices, which also play an important role in proving the claimed identity algebraically. Finding a bijective proof remains an open problem.