This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A337883 #5 Sep 28 2020 21:40:53
%S A337883 1,2,1,3,5,1,4,15,40,1,5,36,1197,3504,1,6,75,18592,9753615,13724608,1,
%T A337883 7,141,166885,3056311808,19854224207910,3574466244480,1,8,245,1019880,
%U A337883 264940140875,468488921670219776,25959704193068472575379,106607224611810055168,1
%N A337883 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 A337883 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 A337883 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 A337883 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 A337883 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 A337883 T(n,k) = A337884(n,k) + A337885(n,k) = 2*A337884(n,k) - A337886(n,k) = 2*A337885(n,k) + A337886(n,k).
%e A337883 The table begins with T(2,1):
%e A337883 1 2 3 4 5 6 7 ...
%e A337883 1 5 15 36 75 141 245 ...
%e A337883 1 40 1197 18592 166885 1019880 4738153 ...
%e A337883 1 3504 9753615 3056311808 264940140875 10156268150064 221646915632373 ...
%e A337883 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 A337883 m=2; (* dimension of color element, here a triangular face *)
%t A337883 lw[n_,k_]:=lw[n, k]=DivisorSum[GCD[n,k],MoebiusMu[#]Binomial[n/#,k/#]&]/n (*A051168*)
%t A337883 cxx[{a_, b_},{c_, d_}]:={LCM[a, c], GCD[a, c] b d}
%t A337883 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 A337883 combine[a : {{_, _} ...}, b : {{_, _} ...}] := Outer[cxx, a, b, 1]
%t A337883 CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)
%t A337883 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 A337883 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 A337883 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A337883 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 A337883 array[n_, k_] := row[n] /. j -> k
%t A337883 Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten
%Y A337883 Cf. A337884 (unoriented), A337885 (chiral), A337886 (achiral), A051168 (binary Lyndon words).
%Y A337883 Other elements: A324999 (vertices), A327083 (edges).
%Y A337883 Other polytopes: A337887 (orthotope), A337891 (orthoplex).
%Y A337883 Rows 2-4 are A000027, A006008, A331350.
%K A337883 nonn,tabl
%O A337883 2,2
%A A337883 _Robert A. Russell_, Sep 28 2020