A385390 Irregular triangle read by rows: T(n,k) is the number of polysticks of size k, i.e., connected subsets of k edges, 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 <= 2*n^2.

1, 1, 1, 2, 3, 7, 4, 4, 1, 1, 1, 2, 5, 14, 38, 111, 261, 500, 654, 648, 486, 305, 144, 61, 19, 6, 1, 1, 1, 2, 5, 16, 52, 199, 759, 2921, 10668, 36761, 115231, 322237, 778242, 1576259, 2591721, 3412285, 3671098, 3320276, 2565917, 1717088, 996355, 503860, 220074, 83408, 26783, 7438, 1678, 351, 52, 11, 1, 1

Offset: 1

Author

Pontus von Brömssen, Jun 27 2025

Keywords

• nonn
• tabf

Comments

For n = 4, there are 384 automorphisms of (the line graph 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 A333333.

Examples

			Triangle begins:
  1, 1;
  1, 2, 3,  7,  4,   4,   1,   1;
  1, 2, 5, 14, 38, 111, 261, 500, 654, 648, 486, 305, 144, 61, 19, 6, 1, 1;
  ...
		

Crossrefs

Cf. A019988, A333333, A385385 (polyominoes), A385388 (interchange of rows and columns of the torus not allowed), A385389 (row sums).

Formula

T(n,k) = A019988(k) if n >= k.

T(n,k) >= A385388(n,k)/2, with equality if and only if k is odd.