Advances in Mathematics· 2026Q1
Diagonally symmetric alternating sign matrices
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- Q1SCImago
- 2026year
Short summary
A Pfaffian formula precisely counts diagonally symmetric alternating sign matrices (DSASMs) of any size, using binomial coefficients, and also enumerates DSASMs based on specific entry counts and positions.
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Key points
- A Pfaffian formula is derived for the exact enumeration of diagonally symmetric alternating sign matrices (DSASMs).
- The formula's entries are positive integers expressed as simple binomial coefficient combinations.
- Pfaffian formulas are also presented for generating functions related to specific DSASM statistics (nonzero entries, diagonal entries, first-row '1' position).
- The proofs utilize a bijection between DSASMs and a six-vertex model configuration.
AI-generated from the title and abstract; the full text is not read.
Abstract
The enumeration of diagonally symmetric alternating sign matrices (DSASMs) is studied, and a Pfaffian formula is obtained for the number of DSASMs of any fixed size, where the entries for the Pfaffian are positive integers given by simple binomial coefficient expressions. This result provides the first known case of an exact enumeration formula for an alternating sign matrix symmetry class in which a simple product formula does not seem to exist. Pfaffian formulae are also obtained for DSASM generating functions associated with several natural statistics, including the number of nonzero strictly upper triangular entries in a DSASM, the number of nonzero diagonal entries in a DSASM, and the column number of the unique 1 in the first row of a DSASM. The proofs of these results involve introducing a version of the six-vertex model whose configurations are in bijection with DSASMs of fixed size, and obtaining a Pfaffian expression for its partition function. Various related topics are also studied, involving diagonally symmetric permutation matrices, off-diagonally symmetric alternating sign matrices, certain natural involutions on DSASMs, and the asymptotic enumeration of DSASMs and other classes of alternating sign matrices.
The authors' abstract, as published at the source. Advances in Mathematics, 2026 · DOI ↗
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Field: Statistics and Probability
Statistics and ProbabilityMathematics