A bijective proof of the ASM theorem, Part I: the operator formula

Ilse Fischer; Matjaz Konvalinka

A bijective proof of the ASM theorem, Part I: the operator formula

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arxiv 1910.04198 v1 pith:XK73J37A submitted 2019-10-09 math.CO

Ilse Fischer , Matjaz Konvalinka This is my paper · ORCID

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keywords proofbijectivealternatingbeenequivalencesformulaoperatorpartitions

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Alternating sign matrices are known to be equinumerous with descending plane partitions, totally symmetric self-complementary plane partitions and alternating sign triangles, but no bijective proof for any of these equivalences has been found so far. In this paper we provide the first bijective proof of the operator formula for monotone triangles, which has been the main tool for several non-combinatorial proofs of such equivalences. In this proof, signed sets and sijections (signed bijections) play a fundamental role.

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Frozen-corner enumeration of Alternating Sign Matrices
math.CO 2025-09 conditional novelty 3.0 of 10

The number of ASMs with an s by s frozen zero corner is conjectured to equal A_n det(1-M), a determinant formula verified numerically for all n up to 20.

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