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A327083 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the edges of a regular n-dimensional simplex using up to k colors. 9
1, 2, 1, 3, 4, 1, 4, 11, 12, 1, 5, 24, 87, 40, 1, 6, 45, 416, 1197, 184, 1, 7, 76, 1475, 18592, 42660, 1296, 1, 8, 119, 4236, 166885, 3017600, 4223313, 17072, 1, 9, 176, 10437, 1019880, 85025050, 1748176768, 1139277096, 424992 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.

A(n,k) is also the number of oriented colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using up to k colors. Thus, A(2,k) also counts the number of oriented colorings of the vertices (0-dimensional simplices) of an equilateral triangle.

LINKS

Robert A. Russell, Table of n, a(n) for n = 1..325 First 25 antidiagonals.

Harald Fripertinger, The cycle type of the induced action on 2-subsets

E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

FORMULA

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.

A(n,k) = Sum_{j=1..(n+1)*n/2} A327087(n,j) * binomial(k,j).

A(n,k) = A327084(n,k) + A327085(n,k) = 2*A327084(n,k) - A327086(n,k) = 2*A327085(n,k) + A327086(n,k).

EXAMPLE

Array begins with A(1,1):

1  2    3     4      5       6       7        8        9        10        11

1  4   11    24     45      76     119      176      249       340       451

1 12   87   416   1475    4236   10437    22912    45981     85900    151371

1 40 1197 18592 166885 1019880 4738153 17962624 58248153 166920040 432738229

For A(2,3) = 11, the nine achiral colorings are AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, and CCC. The chiral pair is ABC-ACB.

MATHEMATICA

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

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

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)

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]]]

pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (*partition count*)

row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#, 2]]], pc[#] j^Total[CycleX[#]][[2]], 0] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]

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

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

(* Using Fripertinger's exponent per Andrew Howroyd's code in A063841: *)

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

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

array[n_, k_] := Total[If[EvenQ[Total[1-Mod[#, 2]]], pc[#]k^ex[#], 0] &/@ IntegerPartitions[n+1]]/((n+1)!/2)

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

CROSSREFS

Cf. A327084 (unoriented), A327085 (chiral), A327086 (achiral), A327087 (exactly k colors), A324999 (vertices).

Rows 1-3 are A000027, A006527, A046023.

Column 2 is A218144(n+1).

Sequence in context: A137649 A180915 A240783 * A104002 A073135 A063804

Adjacent sequences:  A327080 A327081 A327082 * A327084 A327085 A327086

KEYWORD

nonn,tabl

AUTHOR

Robert A. Russell, Aug 19 2019

STATUS

approved

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Last modified August 7 11:34 EDT 2020. Contains 336275 sequences. (Running on oeis4.)