# Patterns of alternating sign matrices

@article{Brualdi2011PatternsOA, title={Patterns of alternating sign matrices}, author={Richard A. Brualdi and Kathleen Kiernan and Seth A. Meyer and Michael W. Schroeder}, journal={Linear Algebra and its Applications}, year={2011}, volume={438}, pages={3967-3990} }

Abstract We initiate a study of the zero–nonzero patterns of n × n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without… Expand

#### 24 Citations

Alternating sign and sign-restricted matrices: representations and partial orders

- Mathematics
- The Electronic Journal of Linear Algebra
- 2021

Sign-restricted matrices (SRMs) are $(0, \pm 1)$-matrices where, ignoring 0's, the signs in each column alternate beginning with a $+1$ and all partial row sums are nonnegative. The most investigated… Expand

Alternating sign matrices, extensions and related cones

- Mathematics, Computer Science
- Adv. Appl. Math.
- 2017

The convex cone generated by ASMs of order n, called the ASM cone, is studied, as well as several related cones and polytopes, and the notion of ( 1 ) -doubly stochastic matrices and a generalization of ASMs are introduced and various properties are shown. Expand

Symmetric alternating sign matrices

- Physics, Computer Science
- Australas. J Comb.
- 2014

It is proved that any n×n symmetric (0,−1)-matrix that can be completed to an alternating sign matrix by replacing some 0s with +1s can beCompleted to a symmetric alternating sign matrices. Expand

Alternating sign matrices and hypermatrices, and a generalization of Latin squares

- Mathematics, Computer Science
- Adv. Appl. Math.
- 2018

This work investigates completion problems, in which one asks if a subhypermatrix can be completed (extended) into an ASHM, and shows several theorems of this type of hypermatrix. Expand

Ranks of dense alternating sign matrices and their sign patterns

- Mathematics
- 2015

Abstract In this paper, an explicit formula for the ranks of dense alternating sign matrices is obtained. The minimum rank and the maximum rank of the sign pattern of a dense alternating sign matrix… Expand

Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

- Mathematics, Computer Science
- Graphs Comb.
- 2020

It is shown that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and some of the properties of the resulting sequence of numbers are investigated. Expand

Some Results on Generalized Complementary Basic Matrices and Dense Alternating Sign Matrices

- Mathematics
- 2016

The first part of this dissertation answers the questions posed in the article “A note on permanents and generalized complementary basic matrices”, Linear Algebra Appl. 436 (2012), by M. Fiedler and… Expand

Note on the spectral radius of alternating sign matrices

- Mathematics
- 2014

Abstract We show that the n × n so-called diamond alternating sign matrix D n is the unique n × n alternating sign matrix with maximum spectral radius ρ n , and that lim n → ∞ ρ n n = 2 π .

GENERALIZED ALTERNATING SIGN MATRICES AND SIGNED PERMUTATION MATRICES

- 2021

We continue the investigations in [6] extending the Bruhat order on n× n alternating sign matrices to our more general setting. We show that the resulting partially ordered set is a graded lattice… Expand

Inverses and eigenvalues of diamondalternating sign matrices

- Mathematics
- 2014

Abstract An n × n diamond alternating sign matrix (ASM) is a (0, +1, −1)-matrix with ±1 entries alternatingand arranged in a diamond-shaped pattern. The explicit inverse (for n even) or generalized… Expand

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