A385383 -id:A385383 - cp's OEIS frontend

A385382 Number of polyominoes, i.e., connected nonempty subsets of square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns.

1, 5, 28, 801, 129550

Offset: 1

Pontus von Brömssen, Jun 27 2025

Pontus von Brömssen, Illustration for n = 3.

Main diagonal of A385381.
Row sums of A385383.
Cf. A385384 (interchange of rows and columns of the torus allowed), A385387 (edge subsets).

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.

Original entry on oeis.org

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

Pontus von Brömssen, Jun 27 2025

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.

			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;
  ...

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

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

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

A385388 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; 1 <= k <= 2*n^2.

Original entry on oeis.org

2, 1, 2, 3, 6, 11, 8, 7, 2, 1, 2, 3, 10, 24, 76, 213, 522, 982, 1308, 1274, 972, 593, 288, 114, 38, 10, 2, 1, 2, 3, 10, 28, 104, 387, 1518, 5799, 21336, 73400, 230462, 644155, 1556484, 3151899, 5183442, 6823550, 7342196, 6639409, 5131834, 3433229, 1992710, 1007190, 440148, 166572, 53566, 14806, 3356, 682, 104, 20, 2, 1

Offset: 1

Pontus von Brömssen, Jun 27 2025

			Triangle begins:
  2, 1;
  2, 3,  6, 11,  8,   7,   2,   1;
  2, 3, 10, 24, 76, 213, 522, 982, 1308, 1274, 972, 593, 288, 114, 38, 10, 2, 1;
  ...

Cf. A385383 (polyominoes), A385387 (row sums), A385390 (interchange of rows and columns of the torus allowed).

T(n,k) <= 2*A385390(n,k), with equality if and only if k is odd.

cp's OEIS Frontend

A385382 Number of polyominoes, i.e., connected nonempty subsets of square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns.

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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.

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A385388 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; 1 <= k <= 2*n^2.

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Author

Keywords

Examples

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