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PreprintCC BY-NC-ND 4.0arXiv

A note on near alternating sign matrices with prescribed row and column weights

  • Tommaso Traetta
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Abstract

We study near alternating sign matrices with prescribed row and column weights. After recalling the basic definitions and the necessary conditions coming from the Gale--Ryser theorem, we give a graph-theoretic characterization of the existence problem. More precisely, we show that a near alternating sign matrix with prescribed row and column weight sequences exists if and only if there exists a binary array with the same weights whose associated adjacency graph is bipartite. This reformulation shows that the Gale--Ryser conditions are not sufficient in general. We then formulate a natural conjecture up to permutations of the row and column weight sequences, and present some positive results, based on convex binary arrays and composition constructions.

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Cite

BibTeXRIS
BibTeX
@misc{traetta2026note,
  title = {{A note on near alternating sign matrices with prescribed row and column weights}},
  author = {Tommaso Traetta},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07118},
  eprint = {2609.07118},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07118},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - A note on near alternating sign matrices with prescribed row and column weights
AU  - Tommaso Traetta
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07118
UR  - https://arxiv.org/abs/2609.07118
AB  - We study near alternating sign matrices with prescribed row and column weights. After recalling the basic definitions and the necessary conditions coming from the Gale--Ryser theorem, we give a graph-theoretic characterization of the existence problem. More precisely, we show that a near alternating sign matrix with prescribed row and column weight sequences exists if and only if there exists a binary array with the same weights whose associated adjacency graph is bipartite. This reformulation shows that the Gale--Ryser conditions are not sufficient in general. We then formulate a natural conjecture up to permutations of the row and column weight sequences, and present some positive results, based on convex binary arrays and composition constructions.
ER  -