A385383 Irregular triangle read by rows: T(n,k) is the number of polyominoes of size k, i.e., connected subsets of k square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns; 1 <= k <= n^2.

1, 1, 2, 1, 1, 1, 2, 3, 5, 6, 6, 3, 1, 1, 1, 2, 3, 9, 17, 44, 81, 150, 163, 161, 88, 56, 16, 8, 1, 1, 1, 2, 3, 9, 21, 62, 168, 490, 1324, 3370, 7433, 13905, 20961, 24927, 23008, 16766, 9825, 4669, 1831, 576, 157, 32, 8, 1, 1

Offset: 1

Author

Pontus von Brömssen, Jun 27 2025

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  1, 2, 1, 1;
  1, 2, 3, 5,  6,  6,  3,   1,   1;
  1, 2, 3, 9, 17, 44, 81, 150, 163, 161, 88, 56, 16, 8, 1, 1;
  ...
		

Links

• Pontus von Brömssen, Illustration for row n = 3.

Crossrefs

Cf. A056780, A385382 (row sums), A385385 (interchange of rows and columns of the torus allowed), A385388 (edge subsets).

Formula

T(n,k) = A056780(k) if n >= k.

T(n,k) <= 2*A385385(n,k), with equality if and only if k = 2.