%I #38 Jul 10 2024 02:59:40

%S 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 25506,1044,1,8,84,2451,90005,1601952,2302938,12346,1,9,120,5831,

%U 533358,43571400,892341888,591901884,274668

%N 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 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 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 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 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 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 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 A(n,k) = Sum_{j=1..(n+1)*n/2} A327088(n,j) * binomial(k,j).

%F A(n,k) = A327083(n,k) - A327085(n,k) = (A327083(n,k) + A327086(n,k)) / 2 = A327085(n,k) + A327086(n,k).

%F A(n,k) = A063841(n+1,k-1).

%e Array begins with A(1,1):

%e 1 2 3 4 5 6 7 8 9 10 ...

%e 1 4 10 20 35 56 84 120 165 220 ...

%e 1 11 66 276 900 2451 5831 12496 24651 45475 ...

%e 1 34 792 10688 90005 533358 2437848 9156288 29522961 84293770 ...

%e ...

%e 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 CycleX[{2}] = {{1,1}}; (* cycle index for permutation with given cycle structure *)

%t CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2,1}, {n,(n-2)/2}}, {{n,(n-1)/2}}]

%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 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 pc[p_List] := 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[pc[#] j^Total[CycleX[#]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]

%t array[n_, k_] := row[n] /. j -> k

%t Table[array[n,d-n+1], {d,1,10}, {n,1,d}] // Flatten

%t (* Using Fripertinger's exponent per Andrew Howroyd code in A063841: *)

%t pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]

%t ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i,2,Length[v]}, {j,i-1}] + Total[Quotient[v,2]]

%t array[n_,k_] := Total[pc[#]k^ex[#] &/@ IntegerPartitions[n+1]]/(n+1)!

%t Table[array[n,d-n+1], {d,10}, {n,d}] // Flatten

%t (* Another program (translated from _Andrew Howroyd_'s PARI code): *)

%t 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 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i-1}] + Total[Quotient[v, 2]];

%t T[n_, k_] := Module[{s = 0}, Do[s += permcount[p]*k^edges[p], {p, IntegerPartitions[n+1]}]; s/(n+1)!];

%t Table[T[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jan 08 2021 *)

%o (PARI)

%o 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 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}

%o 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 (Python)

%o from itertools import combinations

%o from math import prod, gcd, factorial

%o from fractions import Fraction

%o from sympy.utilities.iterables import partitions

%o 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 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 Rows 1-4 are A000027, A000292, A063842(n-1), A063843.

%Y Cf. A063841 (k-multigraphs on n nodes).

%K nonn,tabl

%O 1,2

%A _Robert A. Russell_, Aug 19 2019