A385385 -id:A385385 - cp's OEIS frontend

A385384 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, as well as interchange of rows and columns.

Original entry on oeis.org

1, 4, 19, 437, 65325

Offset: 1

Pontus von Brömssen, Jun 27 2025

Row sums of A385385.

Cf. A385382 (interchange of rows and columns of the torus not allowed), A385389 (edge subsets).

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.

Original entry on oeis.org

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

Pontus von Brömssen, Jun 27 2025

			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;
  ...

Pontus von Brömssen, Illustration for row n = 3.

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

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

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

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.

Original entry on oeis.org

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

Pontus von Brömssen, Jun 27 2025

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.

			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;
  ...

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

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

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

cp's OEIS Frontend

A385384 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, as well as interchange of rows and columns.

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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.

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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.

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Examples

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