A338958 Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

68774446614978208476646592, 5523164445430504871588714239322107782006441, 5448873034167734394145221152621861950913444709790439644, 10956401434158576570935650756489255491646473924447332613392130825

Offset: 2

Author

Robert A. Russell, Nov 17 2020

Keywords

• fini
• nonn
• full

Comments

Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

Links

• Robert A. Russell, Table of n, a(n) for n = 2..96

Crossrefs

Cf. A338956 (oriented), A338957 (unoriented), A338959 (achiral), A338954 (up to n colors), A338950 (vertices, facets), A331352 (5-cell), A331360 (8-cell edges, 16-cell faces), A331356 (16-cell edges, 8-cell faces), A338982 (120-cell, 600-cell).

Programs

Mathematica

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (*binomial series*)
Drop[CoefficientList[bp[8]/12+bp[12]/8-bp[16]/24-bp[18]/18-bp[20]/6-5bp[24]/96+bp[32]/24+bp[36]/36-5bp[48]/1152+bp[50]/16-bp[52]/96-bp[60]/96+bp[96]/1152,x],2]

Formula

A338954(n) = Sum_{j=2..Min(n,96)} a(n) * binomial(n,j).

a(n) = A338956(n) - A338957(n) = (A338956(n) - A338959(n)) / 2 = A338957(n) - A338959(n).