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%I A337884 #4 Sep 28 2020 21:41:00
%S A337884 1,2,1,3,5,1,4,15,34,1,5,35,792,2136,1,6,70,10688,4977909,7013320,1,7,
%T A337884 126,90005,1533771392,9930666709494,1788782616656,1,8,210,533358,
%U A337884 132597435125,234249157811872000,12979877431438089379035,53304527811667897248,1
%N A337884 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the triangular faces of a regular n-dimensional simplex using k or fewer colors.
%C A337884 Each chiral pair is counted as one when enumerating unoriented 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 A337884 Also the number of unoriented colorings of the peaks of a regular n-dimensional simplex. A peak of an n-simplex is an (n-3)-dimensional simplex.
%H A337884 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 A337884 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 A337884 T(n,k) = A337883(n,k) - A337885(n,k) = (A337883(n,k) + A337886(n,k)) / 2 = A337885(n,k) + A337886(n,k).
%e A337884 Table begins with T(2,1):
%e A337884 1 2 3 4 5 6 7 ...
%e A337884 1 5 15 35 70 126 210 ...
%e A337884 1 34 792 10688 90005 533358 2437848 ...
%e A337884 1 2136 4977909 1533771392 132597435125 5079767935320 110837593383153 ...
%e A337884 For T(3,4)=35, 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 A337884 m=2; (* dimension of color element, here a triangular face *)
%t A337884 lw[n_,k_]:=lw[n, k]=DivisorSum[GCD[n,k],MoebiusMu[#]Binomial[n/#,k/#]&]/n (*A051168*)
%t A337884 cxx[{a_, b_},{c_, d_}]:={LCM[a, c], GCD[a, c] b d}
%t A337884 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 A337884 combine[a : {{_, _} ...}, b : {{_, _} ...}] := Outer[cxx, a, b, 1]
%t A337884 CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)
%t A337884 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 A337884 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 A337884 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A337884 row[n_Integer] := row[n] = Factor[Total[pc[#] j^Total[CX[#, m+1]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]
%t A337884 array[n_, k_] := row[n] /. j -> k
%t A337884 Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten
%Y A337884 Cf. A337883 (oriented), A337885 (chiral), A337886 (achiral), A051168 (binary Lyndon words).
%Y A337884 Other elements: A325000 (vertices), A327084 (edges).
%Y A337884 Other polytopes: A337888 (orthotope), A337892 (orthoplex).
%Y A337884 Rows 2-4 are A000027, A000332(n+3), A063843.
%K A337884 nonn,tabl
%O A337884 2,2
%A A337884 _Robert A. Russell_, Sep 28 2020