Roger Behrend ; Ilse Fischer ; Matjaz Konvalinka
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Diagonally and antidiagonally symmetric alternating sign matrices of odd order
dmtcs:6346 -
Discrete Mathematics & Theoretical Computer Science,
April 22, 2020,
DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
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https://doi.org/10.46298/dmtcs.6346
Diagonally and antidiagonally symmetric alternating sign matrices of odd orderArticle
Authors: Roger Behrend 1; Ilse Fischer 2; Matjaz Konvalinka 3
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Roger Behrend;Ilse Fischer;Matjaz Konvalinka
1 School of Mathematics [Cardiff]
2 Fakultät für Mathematik [Wien]
3 Faculty of Mathematics and Physics [Ljubljana]
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DAS- ASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang– Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n + 1) × (2n + 1) DASASMs is∏n (3i)! ,andaconjectureofStroganovfrom2008thattheratiobetweenthenumbersof(2n+1)×(2n+1) i=0 (n+i)! DASASMs with central entry −1 and 1 is n/(n + 1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.
Compact enumeration formulas for generalized partitions; Funder: Austrian Science Fund (FWF); Code: Y 463
Bibliographic References
13 Documents citing this article
Marc Munar;Sebastia Massanet;Daniel Ruiz-Aguilera, 2023, A study on the cardinality of some families of discrete operators through alternating sign matrices, Information sciences, 639, pp. 118571, 10.1016/j.ins.2023.01.040, https://doi.org/10.1016/j.ins.2023.01.040.
Arvind Ayyer;Roger E. Behrend;Ilse Fischer, 2020, Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order, arXiv (Cornell University), 367, pp. 107125, 10.1016/j.aim.2020.107125, https://arxiv.org/abs/1611.03823.
Ilse Fischer;Matjaž Konvalinka, 2020, The mysterious story of square ice, piles of cubes, and bijections, Proceedings of the National Academy of Sciences of the United States of America, 117, 38, pp. 23460-23466, 10.1073/pnas.2005525117, https://doi.org/10.1073/pnas.2005525117.
Richard A. Brualdi;Lei Cao, Springer optimization and its applications, Hankel Tournaments and Special Oriented Graphs, pp. 109-152, 2020, 10.1007/978-3-030-55857-4_5.
Arvind Ayyer;Roger E. Behrend, 2019, Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions, Journal of combinatorial theory. Series A, 165, pp. 78-105, 10.1016/j.jcta.2019.01.001, https://doi.org/10.1016/j.jcta.2019.01.001.