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

0,2

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.

Links

Table of n, a(n) for n=0..22.

Example

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]

Prog

(SageMath) nmax = 20
R.<x> = 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())

Crossrefs

Cf. A135401.

Sequence in context: A284716 A107097 A202837 * A180111 A292183 A006923

Adjacent sequences: A370393 A370394 A370395 * A370397 A370398 A370399

Keyword

nonn

Author

Ludovic Schwob, Feb 17 2024

Status

approved