A366541 Regular triangle read by rows: T(n,k) is the number of k-facets of the bipyramid on an n-simplex base.

2, 4, 4, 5, 9, 6, 6, 14, 16, 8, 7, 20, 30, 25, 10, 8, 27, 50, 55, 36, 12, 9, 35, 77, 105, 91, 49, 14, 10, 44, 112, 182, 196, 140, 64, 16, 11, 54, 156, 294, 378, 336, 204, 81, 18, 12, 65, 210, 450, 672, 714, 540, 285, 100, 20, 13, 77, 275, 660, 1122, 1386, 1254, 825, 385, 121, 22

Offset: 0

Author

Jianing Song, Oct 12 2023

Keywords

• nonn
• tabl
• easy

Comments

The bipyramid on an n-simplex base is the direct sum of an n-simplex and a segment. It can also be seen as two (n+1)-simplices augmented together at their base.

Note that all facets are simplices.

A deltatope is a polytope whose all cells are regular simplices (a priori not necessarily having the same size). Every polygon, being regular or not, is a 2-deltatope by definition. There are 8 convex 3-deltatopes or deltahedra (regular tetrahedron, regular octahedron, regular icosahedron, regular triangular bipyramid, regular pentagonal bipyramid and three others), 5 convex 4-deltatopes (regular 5-cell, regular 16-cell, regular 600-cell, regular tetrahedral bipyramid and regular icosahedral bipyramid) and 3 in dimension d >= 5 (regular d-simplex, regular d-orthoplex and regular bipyramid on a (d-1)-simplex base). Note that the regular orthoplex is the regular bipyramid on a hypercube base. It turns out that all cells of a deltatope are congruent (i.e., having the same size) in all nontrivial dimensions (dimension >= 3). See Dr. Richard Klitzing's answer to the Math Overflow question "4-polytopes with only one kind of regular facet" for dimension 4, and Gjergji Zaimi's answer to the question "Convex deltahedra in higher dimensions" for dimension >= 5.

More generally, a convex polytope whose all cells are regular polytopes of the same kind is either regular or a deltatope. See the article of Roswitha Blind.

The symmetry group of the bipyramid on an n-simplex base, generated by the symmetries of the n-simplex and the vertical reflexion that commute, is S_{n+1} X C_2 (with Coxeter notation [2,3^(n-1)]). See the Math Stack Exchange link.

Examples

			The triangle T(n, k) begins:
n\k   0   1    2    3     4     5     6    7    8    9  10
  0   2
  1   4   4
  2   5   9    6
  3   6  14   16    8
  4   7  20   30   25    10
  5   8  27   50   55    36    12
  6   9  35   77  105    91    49    14
  7  10  44  112  182   196   140    64   16
  8  11  54  156  294   378   336   204   81   18
  9  12  65  210  450   672   714   540  285  100   20
 10  13  77  275  660  1122  1386  1254  825  385  121  22
n = 0: the segment has 2 vertices;
n = 1: the quadrilateral has 4 vertices and 4 sides;
n = 2: the triangular bipyramid has 5 vertices, 9 edges and 6 faces;
n = 3: the tetrahedral bipyramid has 6 vertices, 14 edges, 16 faces and 8 cells.
		

Links

• Jianing Song, Table of n, a(n) for n = 0..5150 (Rows n = 0..100)
• Math Overflow, 4-polytopes with only one kind of regular facet.
• Math Overflow, Convex deltahedra in higher dimensions.
• Mathematics Stack Exchange, Group symmetries of a trigonal bipyramidal molecule.
• Roswitha Blind, Jürgen Tölke and Jörg M. Wills, Konvexe Polytope mit regulären Facetten im R^n (n>=4) (in German), Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978.

Crossrefs

A014410(n+1,k) is the number of k-facets of the n-simplex.

Programs

PARI

T(n,k) = if(k
				

Formula

T(n,k) = 2*(number of k-facets of the (n+1)-simplex) - (number of k-facets of the n-simplex) = 2*binomial(n+2,k+1) - binomial(n+1,k+1) for 0 <= k <= n-1; T(n,n) = 2*(number of n-facets of the (n+1)-simplex) - (2 cells as base) = 2*binomial(n+2,n+1) - 2 = 2*(n+1).