A373027 Number of (0,1)-matrices with sum of entries equal to 2*n or 2*n+1, no zero rows or columns, which are centrally symmetric.

1, 4, 32, 400, 6660, 138576, 3460464, 100821904, 3357239440

Offset

0,2

Comments

a(n) is the number ordered pairs 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) and with conjugate shapes, which are both invariant under Schützenberger involution.

Links

Table of n, a(n) for n=0..8.

Example

The a(2) = 32 matrices with sum of entries equal to 4:
  [1 0 0 0] [0 0 0 1] [1 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0]
  [0 1 0 0] [0 1 0 0] [0 0 1 0] [0 0 1 0] [0 0 0 1] [1 0 0 0]
  [0 0 1 0] [0 0 1 0] [0 1 0 0] [0 1 0 0] [1 0 0 0] [0 0 0 1]
  [0 0 0 1] [1 0 0 0] [0 0 0 1] [1 0 0 0] [0 1 0 0] [0 0 1 0]
.
  [0 1 0 0] [0 0 1 0]  [1 0 0] [0 1 0] [0 1 0] [0 0 1]
  [0 0 0 1] [1 0 0 0]  [0 1 0] [1 0 0] [0 0 1] [0 1 0]
  [1 0 0 0] [0 0 0 1]  [0 1 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1 0] [0 1 0 0]  [0 0 1] [0 1 0] [0 1 0] [1 0 0]
.
  [1 0] [0 1] [1 0] [0 1]  [1]
  [1 0] [0 1] [0 1] [1 0]  [1]
  [0 1] [1 0] [1 0] [0 1]  [1]
  [0 1] [1 0] [0 1] [1 0]  [1]
.
  [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]  [0 1 0]  [1 0] [0 1]
  [0 1 1 0] [1 0 0 1] [1 0 0 1] [0 1 1 0]  [1 0 1]  [1 1] [1 1]
  [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 0 0]  [0 1 0]  [0 1] [1 0]
.
  [1 1 0 0] [0 0 1 1] [1 0 1 0] [0 1 0 1]  [1 1 0] [0 1 1]  [1 1]
  [0 0 1 1] [1 1 0 0] [0 1 0 1] [1 0 1 0]  [0 1 1] [1 1 0]  [1 1]
.
  [1 1 1 1]

Crossrefs

Cf. A275787, A370396.

Sequence in context: A349558 A005263 A325574 * A113131 A367376 A195762

Adjacent sequences: A373024 A373025 A373026 * A373028 A373029 A373030

Keyword

nonn,more

Author

Ludovic Schwob, May 20 2024

Status

approved