login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A325013 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors. 10
1, 3, 1, 6, 6, 1, 10, 21, 22, 1, 15, 55, 267, 402, 1, 21, 120, 1996, 132102, 1228158, 1, 28, 231, 10375, 11756666, 484086357207, 400507806843728, 1, 36, 406, 41406, 405385550, 4805323147589984, 74515759884862073604656433, 527471432057653004017274030725792, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.

Also the number of unoriented colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.

LINKS

Robert A. Russell, Table of n, a(n) for n = 1..78

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

Wikipedia, Cross-polytope

FORMULA

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.

A(n,k) = A325012(n,k) - A325014(n,k) = (A325012(n,k) + A325015(n,k)) / 2 = A325014(n,k) + A325015(n,k).

A(n,k) = Sum_{j=1..2^n} A325017(n,j) * binomial(k,j).

EXAMPLE

Array begins with A(1,1):

1   3      6       10        15         21          28           36 ...

1   6     21       55       120        231         406          666 ...

1  22    267     1996     10375      41406      135877       384112 ...

1 402 132102 11756666 405385550 7416923886 86986719477 735192450952 ...

For A(2,2)=6, two squares have all edges the same color, two have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.

MATHEMATICA

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)

a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)

CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)

CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)

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)

cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};

Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];

CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]

CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]

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

row[n_Integer] := row[n] = Factor[(Total[((CI0[#] + CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])

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

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

CROSSREFS

Cf. A325012 (oriented), A325014 (chiral), A325015 (achiral), A325017 (exactly k colors).

Other n-dimensional polytopes: A325000 (simplex), A325005 (orthotope).

Rows 1-4 are A000217, A002817, A128766, A128767; column 2 is A000616.

Cf. A000048, A001037.

Sequence in context: A127893 A127895 A325005 * A152685 A210287 A116412

Adjacent sequences:  A325010 A325011 A325012 * A325014 A325015 A325016

KEYWORD

nonn,tabl,easy

AUTHOR

Robert A. Russell, May 27 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 20 23:34 EDT 2021. Contains 345313 sequences. (Running on oeis4.)