Alternating Sign Matrices

# 1. Definition

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. We say an $n \times n$ alternating sign matrix is of order $n$.

# 2. Examples

Alternating sign matrices of order $3$:

# 8. Dyck Path Tuples

Applying Mp00007 to each row in an alternating sign matrix, you can construct a tuple of nested dyck paths.

# 9. References

[BCS]   Phillip Biane, Luigi Cantini, and Andrea Sportiello, Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions, http://arxiv.org/pdf/1101.3427v1.pdf .

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

# 10. Sage examples

AlternatingSignMatrices (last edited 2015-12-17 22:06:01 by AaronCrenshaw)