A new proof of the refined alternating sign matrix theorem

Ilse Fischer

A new proof of the refined alternating sign matrix theorem

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arxiv math/0507270 v1 pith:J6YV7Z3I submitted 2005-07-13 math.CO

Ilse Fischer This is my paper · ORCID

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keywords alternatingformulamatricessignnumberproofadditionalternative

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abstract

In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of $n \times n$ alternating sign matrices with a 1 at the top of the $i$-th column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(-1) matrices generalizing alternating sign matrices.

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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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