Abstract
We consider the problem of completing a \((0,-1)\)-matrix to an alternating sign matrix (ASM) by replacing some \(0\)s with \(-1\)s. An algorithm can be given to determine a completion or show that one does not exist. We are concerned primarily with bordered-permutation \((0,-1)\) matrices, defined to be \(n\times n\) \((0,-1)\)-matrices with only \(0\)s in their first and last rows and columns where the \(-1\)s form an \((n-2)\times (n-2)\) permutation matrix. We show that any such matrix can be completed to an ASM and characterize those that have a unique completion.
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References
Bressoud, D.: Proofs and Confirmations: the story of te Alternating Sign Conjecture. Cambridge University Press, Mathematical Association of America, Cambridge (1999)
Brualdi, R.A., Kiernan, K.P., Meyer, S.A., Schroeder, M.W.: Patterns of alternating sign matrices. Linear Algebra Appl. 438, 3967–3990 (2013)
Brualdi, R.A., Ryser, H.J.: Comb. Matrix Theory. Cambridge University Press, Cambridge (1991)
Kalantari, B.: Alternating sign matrices and polynomiography. Electr. J. Comb. 18(2), P24 (2011)
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We are grateful to a referee for a thorough and helpful report.
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Research performed while Hwa Kyung Kim is on leave as an Honorary Fellow at University of Wisconsin-Madison. Hwa Kyung Kim was supported by a Research Grant from Sangmyung University.
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Brualdi, R.A., Kim, H.K. Completions of Alternating Sign Matrices. Graphs and Combinatorics 31, 507–522 (2015). https://doi.org/10.1007/s00373-014-1409-1
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DOI: https://doi.org/10.1007/s00373-014-1409-1
Keywords
- Alternating sign matrix (ASM)
- Completion
- Bordered-permutation matrix
- Bipartite graph
- Perfect matching
- Biadjacency matrix
- Permanent