Possible database queries for alternating sign matrices: search your data / browse all statistics / browse all maps

# 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

[Ze92]   D. Zeilberger, Proof of the alternating sign matrix conjecture, Electronic Journal of Combinatorics 3 (1996), R13.