A385381 -id:A385381 - 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).

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.

A385386 Triangle read by rows: T(n,k) is the number of polysticks, i.e., connected nonempty subsets of edges, of the n X k flat torus, up to cyclic shifts and reflections of rows and columns; 1 <= k <= n.

Original entry on oeis.org

3, 7, 40, 14, 225, 6432, 26, 1768, 255451, 43918404

Offset: 1

Pontus von Brömssen, Jun 27 2025

Two edges are connected if they share a vertex.

Subsets that differ by interchanging rows and columns (when n = k) are considered distinct. For example, there are two distinct polysticks of size 1, one horizontal and one vertical. See A385389 for the analog of the main diagonal of this sequence in the case where such subsets are considered identical.

			Triangle begins:
  n\k|  1    2      3        4
  ---+------------------------
  1  |  3
  2  |  7   40
  3  | 14  225   6432
  4  | 26 1768 255451 43918404

Pontus von Brömssen, Illustration for T(3,2).

Cf. A385381 (polyominoes), A385387 (main diagonal), A385389.

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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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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A385386 Triangle read by rows: T(n,k) is the number of polysticks, i.e., connected nonempty subsets of edges, of the n X k flat torus, up to cyclic shifts and reflections of rows and columns; 1 <= k <= n.

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