Alternating Sign Matrix

An alternating sign matrix is a matrix of 0s, 1s, and s in which the entries in each row or column sum to 1 and the nonzero entries in each row and column alternate in sign. The first few for , 2, ... are shown below:

			(1)
			(2)
			(3)
			(4)

Such matrices satisfy the additional property that s in a row or column must have a "outside" it (i.e., all s are "bordered" by s). The numbers of alternating sign matrices for , 2, ... are given by 1, 2, 7, 42, 429, 7436, 218348, ... (OEIS A005130).

The conjecture that the number of is explicitly given by the formula

(5)

now proven to be true, was known as the alternating sign matrix conjecture. can be expressed in closed form as a complicated function of Barnes G-functions, but additional simplification is likely possible.

A recurrence relation for is given by

(6)

where is the gamma function.

Let be the number of alternating sign matrices with one in the top row occurring in the th position. Then

(7)

The result

(8)

for implies (7) (Mills et al. 1983).

Making a triangular array of the number of with a 1 at the top of column gives

(9)

(OEIS A048601), and taking the ratios of adjacent terms gives the array

(10)

(OEIS A029656 and A029638). The fact that these numerators and denominators are respectively the numbers in the (2, 1)- and (1, 2)-Pascal triangles which are different from 1 is known as the refined alternating sign matrix conjecture.