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).

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

Author

Peter Kagey, May 16 2025

Keywords

• nonn
• tabf

Comments

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

Examples

			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;
		

Links

• Peter Kagey, Illustration of row 3.

Crossrefs

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

Cf. A000217, A002905, A292300.

Formula

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

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

Extensions

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