Sign-restricted matrices of 0's, 1's, and −1's

Richard A Brualdi,Geir Dahl

doi:10.1016/j.laa.2021.01.001

Abstract

We study sign-restricted matrices (SRMs), a class of rectangular (0,±1)-matrices generalizing the alternating sign matrices (ASMs). In an SRM each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative. We determine the maximum number of nonzeros in SRMs and characterize the possible row and column sum vectors. Moreover, a number of results on interchange operations are shown, both for SRMs and, more generally, for (0,±1)-matrices. The Bruhat order on ASMs can be extended to SRMs with the result a distributive lattice. Also, we study polytopes associated with SRMs and some relates decompositions.

Highlights

Let m and n be positive integers and let ∆m,n be the set of all m × n matrices each of whose entries is 0, +1, or −1, that is, (0, ±1)-matrices
sign-restricted matrices (SRMs) with row sum vector R and column sum vector S is denoted by S(R, S), or by Sm,n(R, S) if we want to emphasize the dimensions of R and S
The first column of M contains a −1 no matter how the edges of D are ordered, and so some row will begin with a −1, and M is never an SRM

Summary

Introduction

Let m and n be positive integers and let ∆m,n be the set of all m × n matrices each of whose entries is 0, +1, or −1, that is, (0, ±1)-matrices. A sign-restricted matrix (abbreviated here to SRM) is an m × n (0, ±1)-matrix A such that each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative. The set of all SRMs with row sum vector R and column sum vector S is denoted by S(R, S), or by Sm,n(R, S) if we want to emphasize the dimensions of R and S.

Some basic properties of SRMs

Interchanges

Bruhat order

Polytope and decomposition