A370396 Number of nonnegative integer matrices with sum of entries equal to 2*n or 2*n+1, no zero rows or columns, which are symmetric about both diagonals.

1, 3, 13, 63, 347, 2061, 13219, 89877, 646009, 4866339, 38305573, 313535631, 2661927255, 23367856281, 211680786375, 1974332847177, 18929186519705, 186249976522155, 1878195826349765, 19386702579997095, 204603867473735387, 2205553917952342605, 24261717301000314867

Offset: 0

Author

Ludovic Schwob, Feb 17 2024

Keywords

• nonn

Comments

a(n) is the number of semistandard Young tableaux of size 2*n or 2*n+1 with consecutive entries (i.e., if i is in T, and 1<=j<=i, then j is in T) which are invariant under Schützenberger involution.

Examples

			The a(2) = 13 matrices with sum of entries equal to 4:
  [4]
.
  [2 0] [1 1] [0 2]
  [0 2] [1 1] [2 0]
.
  [1 0 0] [0 0 1] [0 1 0]
  [0 2 0] [0 2 0] [1 0 1]
  [0 0 1] [1 0 0] [0 1 0]
.
  [1 0 0 0] [0 0 0 1] [1 0 0 0]
  [0 1 0 0] [0 1 0 0] [0 0 1 0]
  [0 0 1 0] [0 0 1 0] [0 1 0 0]
  [0 0 0 1] [1 0 0 0] [0 0 0 1]
.
  [0 0 0 1] [0 1 0 0] [0 0 1 0]
  [0 0 1 0] [1 0 0 0] [0 0 0 1]
  [0 1 0 0] [0 0 0 1] [1 0 0 0]
  [1 0 0 0] [0 0 1 0] [0 1 0 0]
		

Crossrefs

Cf. A135401.

Programs

SageMath

nmax = 20
R. = PowerSeriesRing(QQ)
S = [R(1)]
for k in range(nmax+1):
    S.append(sum(S[i]*binomial(k,i)*x^(2*(k-i)) for i in range(k+1))/(1-x^2+O(x^(nmax+1)))^k/(1-x+O(x^(nmax+1)))-S[k])
print(sum(1/(1-x+O(x^(nmax+1)))/(1-x^2+O(x^(nmax+1)))^n*sum(x^(2*(n-k))*factorial(n)/factorial(n-k)/factorial(k-i)/factorial(k-j)/factorial(i+j-k)*S[i]*S[j] for k in range(n+1) for i in range(k+1) for j in range(k-i,k+1)) for n in range(nmax+1)).coefficients())