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A337886 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the triangular faces of a regular n-dimensional simplex using k or fewer colors. 10
1, 2, 1, 3, 5, 1, 4, 15, 28, 1, 5, 34, 387, 768, 1, 6, 65, 2784, 202203, 302032, 1, 7, 111, 13125, 11230976, 7109211078, 3098988832, 1, 8, 175, 46836, 254729375, 9393953524224, 50669807706182691, 1831011525739328, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

An achiral arrangement is identical to its reflection. An n-simplex has n+1 vertices. For n=2, the figure is a triangle with one triangular face. For n=3, the figure is a tetrahedron with 4 triangular faces. For higher n, the number of triangular faces is C(n+1,3).

Also the number of achiral colorings of the peaks of a regular n-dimensional simplex. A peak of an n-simplex is an (n-3)-dimensional simplex.

LINKS

Table of n, a(n) for n=2..37.

E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

FORMULA

The algorithm used in the Mathematica program below assigns each permutation of the vertices to a partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition using a formula for binary Lyndon words. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

T(n,k) = A337884(n,k) - A337883(n,k) = A337883(n,k) - 2*A337885(n,k) = A337884(n,k) - A337885(n,k).

EXAMPLE

Table begins with T(2,1):

1   2      3        4         5          6           7            8 ...

1   5     15       34        65        111         175          260 ...

1  28    387     2784     13125      46836      137543       349952 ...

1 768 202203 11230976 254729375 3267720576 28271133933 183296831488 ...

For T(3,4)=34, the 34 achiral arrangements are AAAA, AAAB, AAAC, AAAD, AABB, AABC, AABD, AACC, AACD, AADD, ABBB, ABBC, ABBD, ABCC, ABDD, ACCC, ACCD, ACDD, ADDD, BBBB, BBBC, BBBD, BBCC, BBCD, BBDD, BCCC, BCCD, BCDD, BDDD, CCCC, CCCD, CCDD, CDDD, and DDDD.

MATHEMATICA

m=2; (* dimension of color element, here a triangular face *)

lw[n_, k_]:=lw[n, k]=DivisorSum[GCD[n, k], MoebiusMu[#]Binomial[n/#, k/#]&]/n (*A051168*)

cxx[{a_, b_}, {c_, d_}]:={LCM[a, c], GCD[a, c] b d}

compress[x:{{_, _} ...}] := (s=Sort[x]; For[i=Length[s], i>1, i-=1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]]+=s[[i, 2]]; s=Delete[s, i], Null]]; s)

combine[a : {{_, _} ...}, b : {{_, _} ...}] := Outer[cxx, a, b, 1]

CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)

CX[{n_Integer}, m_] := If[2m>n, CX[{n}, n-m], CX[{n}, m] = Table[{n/k, lw[n/k, m/k]}, {k, Reverse[Divisors[GCD[n, m]]]}]]

CX[p_List, m_Integer] := CX[p, m] = Module[{v = Total[p], q, r}, If[2 m > v, CX[p, v - m], q = Drop[p, -1]; r = Last[p]; compress[Flatten[Join[{{CX[q, m]}}, Table[combine[CX[q, m - j], CX[{r}, j]], {j, Min[m, r]}]], 2]]]]

pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)

row[n_Integer] := row[n] = Factor[Total[If[OddQ[Total[1-Mod[#, 2]]], pc[#] j^Total[CX[#, m+1]][[2]], 0] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]

array[n_, k_] := row[n] /. j -> k

Table[array[n, d+m-n], {d, 8}, {n, m, d+m-1}] // Flatten

CROSSREFS

Cf. A337883 (oriented), A337884 (unoriented), A337885 (chiral), A051168 (binary Lyndon words).

Other elements: A325001 (vertices), A327086 (edges).

Other polytopes: A337890 (orthotope), A337894 (orthoplex).

Rows 2-4 are A000027, A006003, A331353.

Sequence in context: A110197 A124819 A124019 * A337884 A337883 A202179

Adjacent sequences:  A337883 A337884 A337885 * A337887 A337888 A337889

KEYWORD

nonn,tabl

AUTHOR

Robert A. Russell, Sep 28 2020

STATUS

approved

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Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)