Alternating sign matrix

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Not to be confused with Alternant matrix.
The seven alternating sign matrices of size 3

In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. They are also closely related to the six-vertex model with domain wall boundary conditions from statistical mechanics. They were first defined by William Mills, David Robbins, and Howard Rumsey in the former context.


An example of an alternating sign matrix (that is not also a permutation matrix) is

Puzzle picture


Alternating sign matrix conjecture[edit]

The alternating sign matrix conjecture states that the number of n\times n alternating sign matrices is

\prod_{k=0}^{n-1}\frac{(3k+1)!}{(n+k)!} = \frac{1! 4! 7! \cdots (3n-2)!}{n! (n+1)! \cdots (2n-1)!}.

The first few terms in this sequence for n = 0, 1, 2, 3, … are

1, 1, 2, 7, 42, 429, 7436, 218348, … (sequence A005130 in OEIS).

This conjecture was first proved by Doron Zeilberger in 1992.[1] In 1995, Greg Kuperberg gave a short proof[2] based on the Yang-Baxter equation for the six vertex model with domain wall boundary conditions, that uses a determinant calculation,[3] which solves recurrence relations due to Vladimir Korepin.[4]

Razumov–Stroganov conjecture[edit]

In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model (FPL) and ASMs.[5] This conjecture was proved in 2010 by Cantini and Sportiello.[6]


  1. ^ Zeilberger, Doron, Proof of the alternating sign matrix conjecture, Electronic Journal of Combinatorics 3 (1996), R13.
  2. ^ Kuperberg, Greg, Another proof of the alternating sign matrix conjecture, International Mathematics Research Notes (1996), 139-150.
  3. ^ Determinant formula for the six-vertex model, A. G. Izergin et al. 1992 J. Phys. A: Math. Gen. 25 4315.
  4. ^ V. E. Korepin, Calculation of norms of Bethe wave functions, Comm. Math. Phys. Volume 86, Number 3 (1982), 391-418.
  5. ^ Razumov, A.V., Stroganov Yu.G., Spin chains and combinatorics, Journal of Physics A, 34 (2001), 3185-3190.
  6. ^ L. Cantini and A. Sportiello, Proof of the Razumov-Stroganov conjectureJournal of Combinatorial Theory, Series A, 118 (5), (2011) 1549–1574,

Further reading[edit]

External links[edit]