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).

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

Author

Peter Kagey, May 16 2025

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
		

Links

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

Crossrefs

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