# 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

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#### References

SHOWING 1-8 OF 8 REFERENCES
Alternating Sign Matrices and Descending Plane Partitions
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 1983
This paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented. Expand
Proof of the alternating sign matrix conjecture
The number of matrices whose entries are either $-1,$0, or $1$, and such that in every row and every column the non-zero entries alternate in sign, is proved to be 1-4-2-3-2, as conjectured by Mills, Robbins, and Rumsey. Expand
The Many Faces of Alternating-Sign Matrices
• J. Propp
• Mathematics, Computer Science
• DM-CCG
• 2001
I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-functionExpand
Another proof of the alternating sign matrix conjecture
Author(s): Kuperberg, Greg | Abstract: Robbins conjectured, and Zeilberger recently proved, that there are 1!4!7!...(3n-2)!/n!/(n+1)!/.../(2n-1)! alternating sign matrices of order n. We give a newExpand
Cell biology: The story of i
Claire Ainsworth learns how this way of looking at an individual is feeding into immunology and cancer biology, and some sea squirt genes have mammalian counterparts involved in immunity, and there are parallels with stem-cell biology too. Expand
Proofs and Confirmations: The Story of the Alternating Sign Conjecture
• Math. Association of America, Cambridge University Press, Cambridge
• 1999
The story of 1
• 2, 7, 42, 429, 7436, ..., Math. Intelligencer 13
• 1991