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The First Bijective Proof of the Alternating Sign Matrix Theorem Theorem

Authors Ilse Fischer , Matjaž Konvalinka

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Author Details

Ilse Fischer
  • Faculty of Mathematics, University of Vienna, Austria
Matjaž Konvalinka
  • Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

Funding

  • Fischer, Ilse: Austrian Science Foundation FWF, SFB grant F50.
  • Konvalinka, Matjaž: Slovenian Research Agency (research core funding No. P1-0294).

Cite AsGet BibTex

Ilse Fischer and Matjaž Konvalinka. The First Bijective Proof of the Alternating Sign Matrix Theorem Theorem. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.AofA.2020.12

Abstract

Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but a bijective proof for any of these equivalences has been elusive for almost 40 years. In this extended abstract, we provide a sketch of the first bijective proof of the enumeration formula for alternating sign matrices, and of the fact that alternating sign matrices are equinumerous with descending plane partitions. The bijections are based on the operator formula for the number of monotone triangles due to the first author. The starting point for these constructions were known "computational" proofs, but the combinatorial point of view led to several drastic modifications and simplifications. We also provide computer code where all of our constructions have been implemented.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Enumeration
Keywords
  • enumeration
  • bijective proof
  • alternating sign matrix
  • plane partition

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