Generalized rank functions and quilts of alternating sign matrices

TL;DR

This paper introduces quilts of alternating sign matrices, generalizing classical concepts like rank functions and monotone Boolean functions, and explores their properties and enumeration in specific cases.

Contribution

It defines quilts of alternating sign matrices with respect to two posets, establishing their properties, lattice structure, and enumerative results for special poset configurations.

Findings

Quilts form a distributive lattice with notable properties.

Enumeration is achieved when one poset is an antichain or a chain.

Bounds are provided for the number of quilts with a Boolean lattice as one poset.

Abstract

In this paper, we present new objects, quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with respect to two Boolean lattices. When the two posets are chains, a quilt is equivalent to an alternating sign matrix and its corresponding corner sum matrix. Quilts also generalize the monotone Boolean functions counted by the Dedekind numbers. Quilts form a distributive lattice with many beautiful properties and contain many classical and well-known sublattices, such as the lattice of matroids of a given rank and ground set. While enumerating quilts is hard in general, we prove two major enumerative results, when one of the posets is an antichain and when one of them is a chain. We also give some bounds for the number of quilts when one poset is the Boolean lattice.