A383973 -id:A383973 - cp's OEIS frontend

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

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

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).

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

Original entry on oeis.org

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).

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

A384157 Irregular triangle read by rows: T(n,k) is the number of connected induced k-vertex subgraphs of the hyperoctahedral graph of dimension n >= 1 up to automorphisms of the hyperoctahedral graph; 0 <= k <= 2*n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Peter Kagey and Pontus von Brömssen, May 21 2025

In this sequence, the empty graph is considered to be connected.
There are n!*2^n graph automorphisms of the n-hyperoctahedral graph.
The hyperoctahedral graph is also called the "cocktail party graph," and corresponds to the 1-skeleton of the n-dimensional cross-polytope.
Row 3 corresponds to the number of polyominoes on the faces of a cube up to rotation and reflection of the cube.
More generally, this sequence gives the number of k-celled polyforms whose cells are (n-1)-dimensional facets of the n-dimensional hypercube.
An induced subgraph of the hyperoctahedral graph is completely determined (up to automorphisms of the hyperoctahedral graph) by the number i of pairs of antipodal vertices and the number j of vertices whose antipode is not in the subgraph. The subgraph is disconnected if and only if i=1 and j=0. This implies a close relation to A008967 (which also counts disconnected subgraphs); see formula.

			Triangle begins:
  1 | 1, 1, 0;
  2 | 1, 1, 1, 1, 1;
  3 | 1, 1, 1, 2, 2, 1, 1;
  4 | 1, 1, 1, 2, 3, 2, 2, 1, 1;
  5 | 1, 1, 1, 2, 3, 3, 3, 2, 2, 1, 1;
  6 | 1, 1, 1, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1;
  7 | 1, 1, 1, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1;
  8 | 1, 1, 1, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1;
  ...

Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Wikipedia, Cross-polytope

Cf. A008967 (includes disconnected subgraphs), A369605 (hypercube graph), A383973 (edges).

T(n,k) = A008967(n+4,k) if k != 2; T(n,2) = A008967(n+4,2)-1.

G.f.: 1/((1-y)*(1-x*y)*(1-x^2*y)) - x^2*y/(1-y) - 1.

cp's OEIS Frontend

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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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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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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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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A384157 Irregular triangle read by rows: T(n,k) is the number of connected induced k-vertex subgraphs of the hyperoctahedral graph of dimension n >= 1 up to automorphisms of the hyperoctahedral graph; 0 <= k <= 2*n.

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