A333333 -id:A333333 - cp's OEIS frontend

A384067 Number of edge-connected components of n faces of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

1, 2, 1, 3, 5, 11, 19, 36, 50, 48, 34, 15, 7, 2, 1

Offset: 0

Peter Kagey, May 18 2025

Two faces are connected if they share an edge.
These are "free" polyforms because both rotations and reflections are allowed.
The cuboctahedron is the polyhedral dual of the rhombic dodecahedron.

			a(1) = 2 because the cuboctahedron is not face-transitive, but has two distinct types of faces: triangular faces and square faces.

Peter Kagey, Illustration of a(2)-a(4).
Wikipedia, Cuboctahedron

Cf. A333333 (rhombic dodecahedron, row 3).

Cf. A384067 (cuboctahedron), A384068 (truncated cube), A384069 (truncated octahedron), A384070 (rhombicuboctahedron), A384071 (truncated cuboctahedron), A384072 (snub cube).

A369605 Irregular triangle read by rows: T(n,k) is the number of inequivalent connected induced k-vertex subgraphs of the hypercube graph of dimension n >= 0, 1 <= k <= 2^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 1, 1, 1, 1, 1, 3, 5, 11, 19, 36, 37, 41, 24, 18, 6, 4, 1, 1, 1, 1, 1, 3, 5, 17, 44, 158, 493, 1628, 4670, 12266, 27043, 51018, 79042, 103179, 112219, 105232, 84045, 59021, 35533, 19114, 8769, 3716, 1311, 468, 130, 47, 10, 5, 1, 1

Offset: 0

Pontus von Brömssen, Jan 27 2024

Two subgraphs are equivalent if there is an automorphism of the hypercube graph that takes one to the other.
Two isomorphic subgraphs may both be counted. For example, the path with 5 vertices is an induced subgraph of the 4-dimensional hypercube in two inequivalent ways: one that is contained in a 3-dimensional subcube and one that is not. This implies that T(4,5) > A369997(4,5). (In A369997, the subgraphs are counted up to isomorphism.)
Also, number of free k-celled polycubes in n dimensions, whose width in any coordinate direction is at most 2.
Also, number of k-celled polyominoes whose cells are subsets of the (n-1)-dimensional facets of the n-dimensional cross-polytope (or orthoplex). (See A049540.)
A039754 is the corresponding sequence for not necessarily connected subgraphs.

			Triangle begins:
  1;
  1, 1;
  1, 1, 1, 1;
  1, 1, 1, 3, 2,  3,  1,  1;
  1, 1, 1, 3, 5, 11, 19, 36, 37, 41, 24, 18, 6, 4, 1, 1;
  ...
There are T(3,4) = 3 inequivalent connected induced 4-vertex subgraphs of the 3-cube: four vertices of a 2-dimensional face or three vertices of a face together with a vertex from the opposite face, adjacent to either of two inequivalent vertices from the first face.

Cf. A049540 (main diagonal), A333333 (edge-induced subgraphs).
Different ways of counting induced subgraphs in the hypercube graph (totals or by number of vertices):
            \   Subgraphs   |       All       |   Connected
  Symmetries \              |                 |
  --------------------------+-----------------+----------------
  None                      | A001146/ N/A    | A290758/A369999
  Automorphisms of the cube | A000616/A039754 | A369606/A369605
  Isomorphism               | A369996/A369995 | A369998/A369997
  (The N/A entry corresponds to rows 2^n of Pascal's triangle; A345135 comes close.)

T(n,k) = A049540(k) for k <= n+1.

T(n,k) = A039754(n,k) for k > 2^n-n.

Row 5 from Pontus von Brömssen, May 14 2025

A383802 Number of polyforms with n cells on the faces of a tetrakis hexahedron up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 2, 6, 8, 21, 36, 84, 164, 356, 691, 1361, 2342, 3707, 4830, 5082, 3843, 2128, 798, 248, 50, 12, 1, 1

Offset: 0

Peter Kagey, May 10 2025

These are "free" polyforms.

The tetrakis hexahedron is the polyhedral dual of the truncated octahedron.

Peter Kagey, Example of a(5)=8.
Wikipedia, Tetrakis hexahedron

Cf. A383803 (one-sided).
Octahedral symmetry: A333333 (row 3), A383800, A383802, A383804, A383806.
Icosahedral symmetry: A030135, A030136, A340635, A383490, A383492, A383494, A383496.
Cf. A197465 (tetrakis square tiling).

A383973 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-dimensional cross-polytope up to isometries of the polytope, with 0 <= k <= A046092(n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 11, 21, 28, 24, 18, 9, 4, 1, 1, 1, 1, 2, 7, 22, 82, 292, 876, 2023, 3699, 5587, 7099, 7712, 7129, 5668, 3843, 2234, 1099, 475, 169, 57, 16, 5, 1, 1, 1, 1, 2, 7, 25, 114, 584, 3055

Offset: 1

Peter Kagey, May 16 2025

The cross-polytope is also called an orthoplex or a hyperoctahedron.
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 n!*2^n symmetries of the cross-polytope.

			Triangle begins
 1 | 1;
 2 | 1, 1, 1, 1, 1;
 3 | 1, 1, 2, 5, 11, 21, 28, 24, 18, 9, 4, 1, 1;
 4 | 1, 1, 2, 7, 22, 82, 292, 876, 2023, 3699, 5587, 7099, 7712, 7129, 5668, 3843, 2234, 1099, 475, 169, 57, 16, 5, 1, 1;

Peter Kagey, Illustration of the T(3,3)=5 polysticks on the edges of an octahedron of size 3.
Wikipedia, Cross-polytope
Wikipedia, Polystick

Cf. A046092, A333333 (n-cube), A369605.

A383800 Number of polyforms with n cells on the faces of a triakis octahedron up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 10, 13, 28, 42, 81, 130, 239, 369, 587, 817, 1072, 1170, 1054, 594, 217, 46, 11, 1, 1

Offset: 0

Peter Kagey, May 10 2025

These are "free" polyforms.

The triakis octahedron is the polyhedral dual of the truncated cube.

Wikipedia, Triakis octahedron

Cf. A383801 (one-sided).
Octahedral symmetry: A333333 (row 3), A383800, A383802, A383804, A383806.
Icosahedral symmetry: A030135, A030136, A340635, A383490, A383492, A383494, A383496.

A383804 Number of polyforms with n cells on the faces of a deltoidal icositetrahedron up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 65, 166, 453, 1157, 2849, 6252, 11894, 18183, 21614, 19139, 12966, 6691, 2813, 901, 253, 49, 11, 1, 1

Offset: 0

Peter Kagey, May 10 2025

These are "free" polyforms.

The deltoidal icositetrahedron is the polyhedral dual of the rhombicuboctahedron.

Peter Kagey, Illustration of a(5)=23.
Wikipedia, Deltoidal icositetrahedron

Cf. A383805 (one-sided).
Octahedral symmetry: A333333 (row 3), A383800, A383802, A383804, A383806.
Icosahedral symmetry: A030135, A030136, A340635, A383490, A383492, A383494, A383496.

A383806 Number of polyforms with n cells on the faces of a disdyakis dodecahedron up to rotation and reflection.

Original entry on oeis.org

1, 1, 3, 3, 9, 14, 38, 74, 184, 406, 981, 2262, 5398, 12589, 29700, 69289, 161727, 373879, 858884, 1948493, 4358729, 9560977, 20489431, 42663444, 85863997, 165915428, 305531365, 531313203, 863339197, 1294513104, 1765472012, 2153407639, 2304457468, 2119172241, 1641722694

Offset: 0

Peter Kagey, May 10 2025

These are "free" polyforms.

The disdyakis dodecahedron is the polyhedral dual of the truncated cuboctahedron.

Bert Dobbelaere, Table of n, a(n) for n = 0..48
Peter Kagey, Example of a(4)=9.
Wikipedia, Disdyakis dodecahedron

Cf. A383807 (one-sided).
Octahedral symmetry: A333333 (row 3), A383800, A383802, A383804, A383806.
Icosahedral symmetry: A030135, A030136, A340635, A383490, A383492, A383494, A383496.

More terms from Bert Dobbelaere, Jun 08 2025

A383974 Number of connected subsets of n edges of the icosahedron up to the 120 rotations and reflections of the icosahedron.

Original entry on oeis.org

1, 1, 2, 8, 27, 126, 557, 2503, 10270, 37542, 114926, 283958, 552542, 866843, 1129291, 1250835, 1195298, 993613, 720889, 456329, 251444, 119989, 49269, 17238, 5113, 1257, 262, 46, 8, 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 120 symmetries of the icosahedron.

Peter Kagey, Illustration of a(3).

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383975 (tetrahedron, row 3).

a(11)-a(30) from Bert Dobbelaere, May 25 2025

A383799 Irregular triangle: T(n,k) gives the number of k-polysticks on edges of the n-cube up to rotational symmetries of the n-cube, with 0 <= k <= A001787(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 6, 14, 24, 32, 25, 13, 5, 1, 1, 1, 1, 1, 4, 10, 35, 131, 510, 1932, 7123, 24466, 76829, 214685, 518820, 1050433, 1727591, 2273998, 2446653, 2212119, 1709579, 1143416, 663450, 335186, 146371, 55327, 17767, 4898, 1103, 226, 35, 7, 1, 1

Offset: 1

Peter Kagey, May 10 2025

Row 3 gives the number of polyforms with n cells on the faces of a rhombic dodecahedron up to rotation.

			Table begins:
n\k| 0  1  2  3   4   5    6    7     8     9     10     11      12
---+----------------------------------------------------------------
  1| 1, 1;
  2| 1, 1, 1, 1,  1;
  3| 1, 1, 1, 4,  6, 14,  24,  32,   25,   13,     5,     1,      1;
  4| 1, 1, 1, 4, 10, 35, 131, 510, 1932, 7123, 24466, 76829, 214685, ...

Wikipedia, Hypercube
Wikipedia, Polystick

Cf. A001787, A222186.

Cf. A333333 (analogous with reflectional symmetries allowed).

A222186(n) = Sum_{k=0..n*2^(n-1)} T(n,k) - Sum_{k=0..(n-1)*2^(n-2)} T(n-1,k) for n >= 2. - Pontus von Brömssen, May 12 2025

a(30)-a(40) from Pontus von Brömssen, May 14 2025

a(41)-a(53) from Pontus von Brömssen, Jun 11 2025

A383981 Number of connected subsets of n edges of the rhombic dodecahedron up to the 48 rotations and reflections of the rhombic dodecahedron.

Original entry on oeis.org

1, 1, 3, 5, 16, 39, 127, 357, 1067, 2861, 7071, 14827, 25638, 33730, 33189, 24838, 14954, 7188, 2905, 912, 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 rhombic dodecahedron.

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).

cp's OEIS Frontend

A384067 Number of edge-connected components of n faces of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

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A369605 Irregular triangle read by rows: T(n,k) is the number of inequivalent connected induced k-vertex subgraphs of the hypercube graph of dimension n >= 0, 1 <= k <= 2^n.

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A383802 Number of polyforms with n cells on the faces of a tetrakis hexahedron up to rotation and reflection.

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A383973 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-dimensional cross-polytope up to isometries of the polytope, with 0 <= k <= A046092(n-1).

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A383800 Number of polyforms with n cells on the faces of a triakis octahedron up to rotation and reflection.

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A383804 Number of polyforms with n cells on the faces of a deltoidal icositetrahedron up to rotation and reflection.

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A383806 Number of polyforms with n cells on the faces of a disdyakis dodecahedron up to rotation and reflection.

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A383974 Number of connected subsets of n edges of the icosahedron up to the 120 rotations and reflections of the icosahedron.

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A383799 Irregular triangle: T(n,k) gives the number of k-polysticks on edges of the n-cube up to rotational symmetries of the n-cube, with 0 <= k <= A001787(n).

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A383981 Number of connected subsets of n edges of the rhombic dodecahedron up to the 48 rotations and reflections of the rhombic dodecahedron.

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