A385385 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, as well as interchange of rows and columns; 1 <= k <= n^2.

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 2, 1, 1, 1, 1, 2, 5, 10, 23, 44, 80, 87, 86, 49, 32, 10, 5, 1, 1, 1, 1, 2, 5, 12, 32, 88, 249, 675, 1699, 3747, 6993, 10538, 12531, 11580, 8458, 4975, 2378, 943, 305, 87, 19, 5, 1, 1

Offset: 1

Author

Pontus von Brömssen, Jun 27 2025

Keywords

• nonn
• tabf

Comments

For n = 4, there are 384 automorphisms of the 4 X 4 torus grid graph (it is isomorphic to the 4-dimensional hypercube graph), but here we only consider the subgroup consisting of the 128 symmetries of the 4 X 4 torus. Using the full automorphism group of the torus grid graph would change row 4 to the corresponding row of A369605.

Examples

			Triangle begins:
  1;
  1, 1, 1, 1;
  1, 1, 2, 3,  4,  4,  2,  1,  1;
  1, 1, 2, 5, 10, 23, 44, 80, 87, 86, 49, 32, 10, 5, 1, 1;
  ...
		

Crossrefs

Cf. A000105, A369605, A385383 (interchange of rows and columns of the torus not allowed), A385384 (row sums), A385390 (edge subsets).

Formula

T(n,k) = A000105(k) if n >= k.

T(n,k) >= A385383(n,k)/2, with equality if and only if k = 2.