A366541 - OEIS

%I #21 Oct 22 2023 20:32:47

%S 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,

%T 49,14,10,44,112,182,196,140,64,16,11,54,156,294,378,336,204,81,18,12,

%U 65,210,450,672,714,540,285,100,20,13,77,275,660,1122,1386,1254,825,385,121,22

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

%C 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.

%C Note that all facets are simplices.

%C 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.

%C 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.

%C 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.

%H Jianing Song, <a href="/A366541/b366541.txt">Table of n, a(n) for n = 0..5150</a> (Rows n = 0..100)

%H Math Overflow, <a href="https://mathoverflow.net/q/342224">4-polytopes with only one kind of regular facet</a>.

%H Math Overflow, <a href="https://mathoverflow.net/q/149185">Convex deltahedra in higher dimensions</a>.

%H Math Stack Exchange, <a href="https://math.stackexchange.com/q/4059195">Group symmetries of a trigonal bipyramidal molecule</a>.

%H Roswitha Blind, Jürgen Tölke and Jörg M. Wills, <a href="https://doi.org/10.1007/978-3-0348-5765-9_10">Konvexe Polytope mit regulären Facetten im R^n (n>=4)</a> (in German), Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978.

%F 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).

%e The triangle T(n, k) begins:

%e n\k   0   1    2    3     4     5     6    7    8    9  10

%e   0   2

%e   1   4   4

%e   2   5   9    6

%e   3   6  14   16    8

%e   4   7  20   30   25    10

%e   5   8  27   50   55    36    12

%e   6   9  35   77  105    91    49    14

%e   7  10  44  112  182   196   140    64   16

%e   8  11  54  156  294   378   336   204   81   18

%e   9  12  65  210  450   672   714   540  285  100   20

%e  10  13  77  275  660  1122  1386  1254  825  385  121  22

%e n = 0: the segment has 2 vertices;

%e n = 1: the quadrilateral has 4 vertices and 4 sides;

%e n = 2: the triangular bipyramid has 5 vertices, 9 edges and 6 faces;

%e n = 3: the tetrahedral bipyramid has 6 vertices, 14 edges, 16 faces and 8 cells.

%o (PARI) T(n,k) = if(k<n, 2*binomial(n+2,k+1) - binomial(n+1,k+1), 2*(n+1))

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

%K nonn,tabl,easy

%O 0,1

%A _Jianing Song_, Oct 12 2023