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%I #5 Sep 28 2020 21:40:53
%S 1,2,1,3,5,1,4,15,40,1,5,36,1197,3504,1,6,75,18592,9753615,13724608,1,
%T 7,141,166885,3056311808,19854224207910,3574466244480,1,8,245,1019880,
%U 264940140875,468488921670219776,25959704193068472575379,106607224611810055168,1
%N Array read by descending antidiagonals: T(n,k) is the number of oriented colorings of the triangular faces of a regular n-dimensional simplex using k or fewer colors.
%C Each chiral pair is counted as two when enumerating oriented arrangements. 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).
%C Also the number of oriented colorings of the peaks of a regular n-dimensional simplex. A peak of an n-simplex is an (n-3)-dimensional simplex.
%H E. M. Palmer and R. W. Robinson, <a href="https://doi.org/10.1007/BF02392038">Enumeration under two representations of the wreath product</a>, Acta Math., 131 (1973), 123-143.
%F 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).
%F T(n,k) = A337884(n,k) + A337885(n,k) = 2*A337884(n,k) - A337886(n,k) = 2*A337885(n,k) + A337886(n,k).
%e The table begins with T(2,1):
%e 1 2 3 4 5 6 7 ...
%e 1 5 15 36 75 141 245 ...
%e 1 40 1197 18592 166885 1019880 4738153 ...
%e 1 3504 9753615 3056311808 264940140875 10156268150064 221646915632373 ...
%e For T(3,4)=36, 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. The chiral pair is ABCD-ABDC.
%t m=2; (* dimension of color element, here a triangular face *)
%t lw[n_,k_]:=lw[n, k]=DivisorSum[GCD[n,k],MoebiusMu[#]Binomial[n/#,k/#]&]/n (*A051168*)
%t cxx[{a_, b_},{c_, d_}]:={LCM[a, c], GCD[a, c] b d}
%t 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 combine[a : {{_, _} ...}, b : {{_, _} ...}] := Outer[cxx, a, b, 1]
%t CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)
%t 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 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 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t 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 array[n_, k_] := row[n] /. j -> k
%t Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten
%Y Cf. A337884 (unoriented), A337885 (chiral), A337886 (achiral), A051168 (binary Lyndon words).
%Y Other elements: A324999 (vertices), A327083 (edges).
%Y Other polytopes: A337887 (orthotope), A337891 (orthoplex).
%Y Rows 2-4 are A000027, A006008, A331350.
%K nonn,tabl
%O 2,2
%A _Robert A. Russell_, Sep 28 2020
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