Symmetry Classes of Alternating Sign Matrices

David P. Robbins

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Symmetry Classes of Alternating Sign Matrices

David P. Robbins

Combinatorics · Submitted 5 Aug 2000

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,558 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.56 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.55 · compares with prior results 0.14 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign.	background 1.00	0.06	0.02	asserted 1.00
s1	The 8-element group of symmetries of the square acts in an obvious way on square matrices.	background 1.00	0.05	0.02	asserted 1.00
s2	For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup.	background 0.98	0.08	0.02	hedged possibility 0.82
s3	There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P.	background 0.93	0.21	0.08	asserted 0.99
s4	Stanley suggested the study of those alternating sign matrices in each of these symmetry classes.	background 0.98	0.04	0.04	asserted 0.94
s5	We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class.	result 0.97	0.93	0.66	hedged possibility 0.98
s6	Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions.	result 0.50	0.85	0.20	hedged possibility 0.57

What kind of paper the classifier reports this is: empirical study 0.50 · evidence from simulation or computation 0.77.

What was examined, and what was not

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arXiv listing abstract v1

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full text
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references and citing sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices in each of these symmetry classes. We have found evidence suggesting that for six of the symmetry classes there exist simple product formulas for the number of alternating sign matrices in the class. Moreover the factorizations of certain of their generating functions point to rather startling connections between several of the symmetry classes and cyclically symmetric plane partitions.

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References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Enumeration of pattern-avoiding $(0,1)$-matrices and their symmetry classes
math.CO 2025-10 unverdicted novelty 6.0 of 10

Maximal I_k-avoiding (0,1)-matrices are equinumerous with plane partitions of a certain size, with simple product formulas for their ten symmetry classes and a conceptual extension to skew shapes.
Off-diagonally symmetric domino tilings of the Aztec diamond
math.CO 2023-03 unverdicted novelty 6.0 of 10

The number of off-diagonally symmetric domino tilings of the Aztec diamond equals a Pfaffian of a matrix whose entries obey a simple recurrence.

How the literature uses it

15 arXiv papers cite this in 36 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 15 sentences without a confident label

Uses its data or values 1 paper

In each of these cases, data for relatively small values of $n$ is given by Robbins [ref] , and data for somewhat larger values of $n$ is given by Bousquet-Mélou and Habsieger [ref] , and (for TSASMs with $n$ odd) Liénardy [ref] .

Diagonally symmetric alternating sign matrices · ASM symmetry class enumeration · 0.93

Compares with it 1 paper

Note that [ref] and [ref] are closely related to results of Mills, Robbins and Rumsey [ref] , [ref] , Ciucu and Krattenthaler [ref] , Krattenthaler [ref] , [ref] , and Kuperberg [ref] .

Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions · Partitions of double-staircase shape · 0.85

Cites it as background 11 papers

Exact formulas for the number of ASMs invariant under the action of various subgroups of $D_4$ have been derived in a series of papers [ref] .

Off-diagonally symmetric alternating sign matrices · Introduction · 0.92
This suggestion led to numerous conjectures for the straight and weighted enumeration of such symmetric ASMs, with these conjectures being summarized by Robbins in a preprint written in the mid 1980s, and placed on the arXiv in 2000 [ref] .

Diagonally and antidiagonally symmetric alternating sign matrices of odd order · Introduction · 0.89
The systematic study of symmetric alternating sign matrices, initiated by Stanley [ref] , has become a fruitful but arduous task: Robbins initially conjectured a list of simple enumeration formulae [ref] , the last one thereof has only recently been solved [ref] .

Refined Enumeration of Halved Monotone Triangles and Applications to Vertically Symmetric Alternating Sign Trapezoids · Introduction · 0.98

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