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 A327084 #38 Jul 10 2024 02:59:40
%S A327084 1,2,1,3,4,1,4,10,11,1,5,20,66,34,1,6,35,276,792,156,1,7,56,900,10688,
%T A327084 25506,1044,1,8,84,2451,90005,1601952,2302938,12346,1,9,120,5831,
%U A327084 533358,43571400,892341888,591901884,274668
%N A327084 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the edges of a regular n-dimensional simplex using up to k colors.
%C A327084 An n-dimensional simplex has n+1 vertices and (n+1)*n/2 edges. For n=1, the figure is a line segment with one edge. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with six edges. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.
%C A327084 A(n,k) is also the number of unoriented colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using up to k colors. Thus, A(2,k) is also the number of unoriented colorings of the vertices (0-dimensional simplices) of an equilateral triangle.
%H A327084 Chai Wah Wu, <a href="/A327084/b327084.txt">Table of n, a(n) for n = 1..1528</a> (terms 1..325 from Robert A. Russell)
%H A327084 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html/book/hyl00_42.html">The cycle type of the induced action on 2-subsets</a>
%H A327084 E. M. Palmer and R. W. Robinson, <a href="https://projecteuclid.org/euclid.acta/1485889789">Enumeration under two representations of the wreath product</a>, Acta Math., 131 (1973), 123-143.
%F A327084 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.
%F A327084 A(n,k) = Sum_{j=1..(n+1)*n/2} A327088(n,j) * binomial(k,j).
%F A327084 A(n,k) = A327083(n,k) - A327085(n,k) = (A327083(n,k) + A327086(n,k)) / 2 = A327085(n,k) + A327086(n,k).
%F A327084 A(n,k) = A063841(n+1,k-1).
%e A327084 Array begins with A(1,1):
%e A327084 1 2 3 4 5 6 7 8 9 10 ...
%e A327084 1 4 10 20 35 56 84 120 165 220 ...
%e A327084 1 11 66 276 900 2451 5831 12496 24651 45475 ...
%e A327084 1 34 792 10688 90005 533358 2437848 9156288 29522961 84293770 ...
%e A327084 ...
%e A327084 For A(2,3) = 10, the nine achiral colorings are AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, and CCC. The chiral pair is ABC-ACB.
%t A327084 CycleX[{2}] = {{1,1}}; (* cycle index for permutation with given cycle structure *)
%t A327084 CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2,1}, {n,(n-2)/2}}, {{n,(n-1)/2}}]
%t A327084 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 A327084 CycleX[p_List] := CycleX[p] = compress[Join[CycleX[Drop[p, -1]], If[Last[p] > 1, CycleX[{Last[p]}], ## &[]], If[# == Last[p], {#, Last[p]}, {LCM[#, Last[p]], GCD[#, Last[p]]}] & /@ Drop[p, -1]]]
%t A327084 pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A327084 row[n_Integer] := row[n] = Factor[Total[pc[#] j^Total[CycleX[#]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]
%t A327084 array[n_, k_] := row[n] /. j -> k
%t A327084 Table[array[n,d-n+1], {d,1,10}, {n,1,d}] // Flatten
%t A327084 (* Using Fripertinger's exponent per Andrew Howroyd code in A063841: *)
%t A327084 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]
%t A327084 ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i,2,Length[v]}, {j,i-1}] + Total[Quotient[v,2]]
%t A327084 array[n_,k_] := Total[pc[#]k^ex[#] &/@ IntegerPartitions[n+1]]/(n+1)!
%t A327084 Table[array[n,d-n+1], {d,10}, {n,d}] // Flatten
%t A327084 (* Another program (translated from _Andrew Howroyd_'s PARI code): *)
%t A327084 permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
%t A327084 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i-1}] + Total[Quotient[v, 2]];
%t A327084 T[n_, k_] := Module[{s = 0}, Do[s += permcount[p]*k^edges[p], {p, IntegerPartitions[n+1]}]; s/(n+1)!];
%t A327084 Table[T[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jan 08 2021 *)
%o A327084 (PARI)
%o A327084 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A327084 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o A327084 T(n, k) = {my(s=0); forpart(p=n+1, s+=permcount(p)*k^edges(p)); s/(n+1)!} \\ _Andrew Howroyd_, Sep 06 2019
%o A327084 (Python)
%o A327084 from itertools import combinations
%o A327084 from math import prod, gcd, factorial
%o A327084 from fractions import Fraction
%o A327084 from sympy.utilities.iterables import partitions
%o A327084 def A327084_T(n,k): return int(sum(Fraction(k**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n+1))) # _Chai Wah Wu_, Jul 09 2024
%Y A327084 Cf. A327083 (oriented), A327085 (chiral), A327086 (achiral), A327088 (exactly k colors), A325000 (vertices, facets), A337884 (faces, peaks), A337408 (orthotope edges, orthoplex ridges), A337412 (orthoplex edges, orthotope ridges).
%Y A327084 Rows 1-4 are A000027, A000292, A063842(n-1), A063843.
%Y A327084 Cf. A063841 (k-multigraphs on n nodes).
%K A327084 nonn,tabl
%O A327084 1,2
%A A327084 _Robert A. Russell_, Aug 19 2019