A392029 Number of sparse alternating sign matrices of size n.

1, 1, 2, 7, 40, 364, 5100, 107601, 3341636, 149771253, 9514938920

Offset

0,3

Comments

An alternating sign matrix (ASM) is sparse if it cannot be turned into another ASM by replacing nonzero entries by zeros.

Links

Table of n, a(n) for n=0..10.

Example

The first ASM is not sparse, because it can be turned into the second one by replacing nonzero entries by zeros:
  [0 + 0 0]       [0 + 0 0]
  [+ - + 0]       [+ 0 0 0]
  [0 + - +]       [0 0 0 +]
  [0 0 + 0]       [0 0 + 0]

Prog

(SageMath)
def is_sparse_ASM(M):
    M = M.to_matrix()
    n = M.nrows()
    E = []
    for i in range(1, n-1):
        a1, b1 = 0, None
        a2, b2 = 0, None
        for j in range(1, n):
            if M[i][j] == 1:
                a1 = j
                if b1:
                    E.append(((i, a1), (i, b1)))
            elif M[i][j] == -1:
                b1 = j
                E.append(((i, a1), (i, b1)))
            if M[j][i] == 1:
                a2 = j
                if b2:
                    E.append(((b2, i), (a2, i)))
            elif M[j][i] == -1:
                b2 = j
                E.append(((b2, i), (a2, i)))
    return DiGraph(E).is_directed_acyclic()
a = lambda n: sum(1 for M in AlternatingSignMatrices(n) if is_sparse_ASM(M))
print([a(k) for k in range(7)])

Crossrefs

Cf. A005130.

Sequence in context: A028441 A006455 A130715 * A317723 A392410 A340005

Adjacent sequences: A392026 A392027 A392028 * A392030 A392031 A392032

Keyword

nonn,more

Author

Ludovic Schwob, Feb 22 2026

Extensions

a(8)-a(10) from Christian Sievers, Mar 06 2026

Status

approved