|
| |
|
|
A366541
|
|
Regular triangle read by rows: T(n,k) is the number of k-facets of the bipyramid on an n-simplex base.
|
|
1
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
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.
|
|
|
LINKS
|
|
|
|
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).
|
|
|
EXAMPLE
|
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.
|
|
|
PROG
|
(PARI) T(n, k) = if(k<n, 2*binomial(n+2, k+1) - binomial(n+1, k+1), 2*(n+1))
|
|
|
CROSSREFS
|
A014410(n+1,k) is the number of k-facets of the n-simplex.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
