The mysterious story of square ice, piles of cubes, and bijections | Zendy

Ilse Fischer | Zendy; Matjaž Konvalinka | Zendy

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The mysterious story of square ice, piles of cubes, and bijections

Author(s) -

Ilse Fischer,

Matjaž Konvalinka

Publication year - 2020

Publication title -

proceedings of the national academy of sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 5.011

H-Index - 771

eISSN - 1091-6490

pISSN - 0027-8424

DOI - 10.1073/pnas.2005525117

Subject(s) - bijection, injection and surjection , bijection , plane (geometry) , combinatorics , mathematics , sign (mathematics) , simple (philosophy) , square (algebra) , construct (python library) , computer science , geometry , mathematical analysis , philosophy , epistemology , programming language

Significance In the early 1980s, it was discovered that alternating sign matrices (ASMs), which are also commonly encountered in statistical mechanics, are counted by the same numbers as two classes of plane partitions. Since then it has been a major open problem in this area to construct explicit bijections between the three classes of objects. In this paper we describe a bijective proof that relates ASMs to one of the two classes of plane partitions as well as a bijective proof of the fact that the numbers can be expressed by a simple product formula. The constructions are complicated, but they are completely explicit and also provided in the form of computer code.

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