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1. Definition & Example
An alternating sign matrix (ASM) is a square matrix whose entries all belong to $\{-1,0,1\}$ such that the sum of each row and column is 1 and the non-zero entries in each row and column alternate in sign.
The size of an alternating sign matrix is the size of the square matrix.
The seven alternating sign matrices of size 3 |
||||||
[[1,0,0],[0,1,0],[0,0,1]] |
[[0,1,0],[1,0,0],[0,0,1]] |
[[1,0,0],[0,0,1],[0,1,0]] |
[[0,1,0],[1,-1,1],[0,1,0]] |
[[0,0,1],[1,0,0],[0,1,0]] |
[[0,1,0],[0,0,1],[1,0,0]] |
[[0,0,1],[0,1,0],[1,0,0]] |
The number of alternating sign matrices of size $n$ is $$\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!},$$ see A005130. This result was known as the alternating sign matrix conjecture before being proved in [Ze92].
2. FindStat representation and coverage
An alternating sign matrix is uniquely represented as a list of lists representing its rows.
Alternating sign matrices are graded by its size.
- The database contains all alternating sign matrices of size at most 6.
3. Additional information
ASM's are in bijection with square-ice models (also known as 6-vertex models) that satisfy the domain-wall boundary conditions.
4. References
5. Sage examples