A385389 -id:A385389 - 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).

A385387 Number of polysticks, i.e., connected nonempty subsets of edges, of the n X n flat torus, up to cyclic shifts and reflections of rows and columns.

Original entry on oeis.org

3, 40, 6432, 43918404

Offset: 1

Pontus von Brömssen, Jun 27 2025

Main diagonal of A385386.
Row sums of A385388.
Cf. A385382 (polyominoes), A385389 (interchange of rows and columns of the torus allowed).

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.

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.

Views

Author

Keywords

Crossrefs

A385387 Number of polysticks, i.e., connected nonempty subsets of edges, of the n X n flat torus, up to cyclic shifts and reflections of rows and columns.

Views

Author

Keywords

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula