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A338113 Triangle read by rows: T(n,k) is the number of oriented colorings of the faces (and peaks) of a regular n-dimensional simplex using exactly k colors. Row n has C(n+1,3) columns.

%I A338113 #9 Oct 14 2020 10:37:52
%S A338113 1,1,3,3,2,1,38,1080,14040,85500,274104,493920,504000,272160,60480,1,
%T A338113 3502,9743106,3017318368,249756082950,8612276962188,156010151929968,
%U A338113 1699145259725088,12107373916276800,59649257217110400
%N A338113 Triangle read by rows: T(n,k) is the number of oriented colorings of the faces (and peaks) of a regular n-dimensional simplex using exactly k colors. Row n has C(n+1,3) columns.
%C A338113 An n-dimensional simplex has n+1 vertices, C(n+1,3) faces, and C(n+1,3) peaks, which are (n-3)-dimensional simplexes. For n=2, the figure is a triangle with one face. For n=3, the figure is a tetrahedron with four triangular faces and four peaks (vertices). For n=4, the figure is a 4-simplex with ten triangular faces and ten peaks (edges). The Schläfli symbol {3,...,3}, of the regular n-dimensional simplex consists of n-1 3's. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
%C A338113 The algorithm used in the Mathematica program below assigns each permutation of the vertices to a cycle-structure partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).
%H A338113 G. Royle, <a href="http://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf">Partitions and Permutations</a>
%F A338113 A337883(n,k) = Sum_{j=1..C(n+1,3)} T(n,j) * binomial(k,j).
%F A338113 T(n,k) = A338114(n,k) + A338115(n,k) = 2*A338114(n,k) - A338116(n,k) = 2*A338115(n,k) + A338116(n,k).
%F A338113 T(3,k) = A325002(3,k); T(4,k) = A327087(4,k).
%e A338113 Triangle begins with T(2,1):
%e A338113   1
%e A338113   1  3    3     2
%e A338113   1 38 1080 14040 85500 274104 493920 504000 272160 60480
%e A338113   ...
%e A338113 For T(3,2)=3, the tetrahedron has one, two, or three faces (vertices) of one color.
%t A338113 m=2; (* dimension of color element, here a triangular face *)
%t A338113 lw[n_, k_]:=lw[n, k]=DivisorSum[GCD[n, k], MoebiusMu[#]Binomial[n/#, k/#]&]/n (*A051168*)
%t A338113 cxx[{a_, b_}, {c_, d_}]:={LCM[a, c], GCD[a, c] b d}
%t A338113 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)
%t A338113 combine[a : {{_, _} ...}, b : {{_, _} ...}] := Outer[cxx, a, b, 1]
%t A338113 CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)
%t A338113 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]]]}]]
%t A338113 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]]]]
%t A338113 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A338113 row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#, 2]]], pc[#] j^Total[CX[#, m+1]][[2]], 0] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]
%t A338113 array[n_, k_] := row[n] /. j -> k
%t A338113 Table[LinearSolve[Table[Binomial[i,j],{i,Binomial[n+1,m+1]},{j,Binomial[n+1,m+1]}], Table[array[n,k],{k,Binomial[n+1,m+1]}]], {n,m,m+4}] // Flatten
%Y A338113 Cf. A338114 (unoriented), A338115 (chiral), A338116 (achiral), A337883 (k or fewer colors), A325002 (vertices and facets), A327087 (edges and ridges).
%K A338113 nonn,tabf
%O A338113 2,3
%A A338113 _Robert A. Russell_, Oct 10 2020