A333333 -id:A333333 - cp's OEIS frontend

A383982 Number of connected subsets of n edges of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

1, 1, 3, 7, 24, 74, 269, 876, 2788, 7639, 17828, 32326, 44375, 46456, 39213, 26865, 15470, 7278, 2917, 913, 254, 49, 11, 1, 1

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 48 symmetries of the cuboctahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383983 Number of connected subsets of n edges of the rhombic triacontahedron up to the 120 rotations and reflections of the rhombic triacontahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 84, 334, 1330, 5495, 22776, 94920, 394706

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 120 symmetries of the rhombic triacontahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383984 Number of connected subsets of n edges of the icosidodecahedron up to the 120 rotations and reflections of the icosidodecahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 81, 323, 1265, 5202, 21335, 88412, 364897

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 120 symmetries of the icosidodecahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383825 Number of polyforms with n cells on the faces of a triakis tetrahedron up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 9, 9, 14, 10, 5, 1, 1

Offset: 0

Peter Kagey, May 11 2025

These are "free" polyforms.

The triakis tetrahedron is the polyhedral dual of the truncated tetrahedron.

Peter Kagey, Illustration of a(0)-a(4).
Wikipedia, Triakis tetrahedron

Cf. A383826 (one-sided).
Octahedral symmetry: A333333 (row 3), A383800, A383802, A383804, A383806.
Icosahedral symmetry: A030135, A030136, A340635, A383490, A383492, A383494, A383496.

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.

A333418 Irregular triangle: T(n,k) gives the number of ways to 2-color k edges of the n-cube up to rotation and reflection, with 0 <= k <= A001787(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 9, 18, 24, 30, 24, 18, 9, 4, 1, 1, 1, 1, 6, 24, 140, 604, 2596, 9143

Offset: 1

Peter Kagey, Mar 20 2020

Conjecture: All rows are unimodal (increasing, then decreasing).
Each row is a palindrome.
A333333 is analogous with the restriction that the colorings must be connected.

			Table begins:
n\k| 0  1   2   3    4    5     6     7   8  9 10 11 12 ...
---+-------------------------------------------------------
  1| 1, 1;
  2| 1, 1,  2,  1,   1;
  3| 1, 1,  4,  9,  18,  24,   30,   24, 18, 9, 4, 1, 1;
  4| 1, 1,  6, 24, 140, 604, 2596, 9143, ...
  5| 1, 1,  8, 50, 608, ...
  6| 1, 1, 10, 89, ...

Mathematics Stack Exchange, Number of 2-colorings of edges of the n-dimensional cube?.
Wikipedia, Hypercube

Row sums are A333444.

Cf. A001787, A060530, A199406, A331359, A333444, A333333.

T(n,k) >= ceiling(binomial(A001787(n),k)/A000165(n)).

A383975 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-simplex up to isometries of the n-simplex, with 0 <= k <= A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 3, 5, 6, 6, 4, 2, 1, 1, 1, 1, 1, 3, 5, 12, 19, 23, 24, 21, 15, 9, 5, 2, 1, 1, 1, 1, 1, 3, 5, 12, 30, 56, 91, 128, 147, 147, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1, 1, 1, 1, 3, 5, 12, 30, 79, 180, 364, 633, 961, 1300, 1551, 1644, 1556, 1311, 980, 663, 402, 221, 115, 56, 24, 11, 5, 2, 1, 1

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks", "polyedges", and "polyforms".
These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other via the n! symmetries of the n-simplex.
Equivalently, T(n,k) is the number of connected unlabeled graphs with k edges and between 1 and n+1 vertices. - Pontus von Brömssen, May 27 2025

			Triangle begins:
 0 | 1;
 1 | 1, 1;
 2 | 1, 1, 1, 1;
 3 | 1, 1, 1, 3, 2, 1, 1;
 4 | 1, 1, 1, 3, 5, 6, 6, 4, 2, 1, 1;
 5 | 1, 1, 1, 3, 5, 12, 19, 23, 24, 21, 15, 9, 5, 2, 1, 1;
 6 | 1, 1, 1, 3, 5, 12, 30, 56, 91, 128, 147, 147, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1;

Peter Kagey, Illustration of row 3.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron).

Cf. A000217, A002905, A292300.

T(n,n) = A002905(n).

The sum of row n is A292300(n+1)+1 for n >= 1. - Pontus von Brömssen, May 26 2025

Missing term a(62)=1 inserted and a(73)-a(91) added by Pontus von Brömssen, May 26 2025

A333362 Number of free polysticks with n segments on the edges of the n-cube.

Original entry on oeis.org

1, 1, 3, 7, 27, 121, 751

Offset: 1

Peter Kagey, Mar 16 2020

A free polystick is a polystick counted up to isometries of the n-cube.

Code Golf Stack Exchange, Counting polysticks on the n-cube.
Wikipedia, Hypercube.
Wikipedia, Polystick.

Cf. A019988.

a(n) = A333333(n,n) = A333333(n+k,n) for all k >= 0.

Definition corrected by Peter Kagey, Jun 13 2023

cp's OEIS Frontend

A383982 Number of connected subsets of n edges of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

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A383983 Number of connected subsets of n edges of the rhombic triacontahedron up to the 120 rotations and reflections of the rhombic triacontahedron.

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A383984 Number of connected subsets of n edges of the icosidodecahedron up to the 120 rotations and reflections of the icosidodecahedron.

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A383825 Number of polyforms with n cells on the faces of a triakis tetrahedron up to rotation and reflection.

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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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A333418 Irregular triangle: T(n,k) gives the number of ways to 2-color k edges of the n-cube up to rotation and reflection, with 0 <= k <= A001787(n).

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A383975 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-simplex up to isometries of the n-simplex, with 0 <= k <= A000217(n).

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A333362 Number of free polysticks with n segments on the edges of the n-cube.

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