A144151 -id:A144151 - cp's OEIS frontend

A167986 Triangle T(n,k) = Number of k-cycles on the graph of an n-orthoplex. n>=2, k>=3.

0, 1, 8, 15, 24, 16, 32, 102, 288, 640, 960, 744, 80, 370, 1584, 5920, 18240, 43080, 69120, 56256, 160, 975, 5664, 30080, 141120, 564120, 1835520, 4542336, 7580160, 6385920, 280, 2121, 15624, 108080, 684480, 3876600, 19138560, 79805376

Offset: 2

Andrew Weimholt, Nov 16 2009

Row n contains 2n-2 elements.
The n-orthoplex is the dual polytope of the n-cube.
The orthoplex is also known as the cross-polytope.
Also the triangle of coefficients of the cocktail party graph cycle polynomials ordered from smallest to largest exponent starting with x^3. - Eric W. Weisstein

			T(3,3) = 8, because in dimension n=3, the cross-polytope is the octahedron, which has 8 3-cycles in its graph.
Triangle starts
   0,   1;
   8,  15,   24,   16;
  32, 102,  288,  640,   960,   744;
  80, 370, 1584, 5920, 18240, 43080, 69120, 56256;
  ...
In terms of cycle polynomials:
   0*x^3 +    1*x^4;
   8*x^3 +  15*x^4 +  24*x^5 +  16*x^6;
  32*x^3 + 102*x^4 + 288*x^5 + 640*x^6 + 960*x^7 + 744*x^8;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..871
Eric Weisstein's World of Mathematics, Cross Polytope
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167987 (row sums).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (k-cycles on (n-1)-simplex for k>3).

b:= func< n,k,j | (-1)^j*Binomial(n,j)*Binomial(2*(n-j),k-2*j)*2^(j-1)*Factorial(k-j-1) >;
A167986:= func< n,k | (&+[b(n,k,j): j in [0..Floor(k/2)]]) >;
[A167986(n,k): k in [3..2*n], n in [2..10]]; // G. C. Greubel, Jan 17 2023

T[n_, k_]:= Sum[(-1)^j*Binomial[n, j]*Binomial[2*(n-j), k-2*j]*2^j*(k - j - 1)!/2, {j, 0, Floor[k/2]}];
Table[T[n, k], {n,2,7}, {k,3,2*n}]//Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Binomial[2n, k]*Gamma[k]*HypergeometricPFQ[{(1-k)/2, -k/2}, {1 - k, 1/2 -n}, -2]/2, {n,7}, {k,3,2n}]//Flatten (* Eric W. Weisstein, Mar 25 2020 *)

a(n,k)=sum(j=0,k\2, (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!)/2;
for (n=2,6,for (k=3,2*n, print1(a(n,k), ","));print); \\ Andrew Howroyd, May 09 2017

def A167986(n,k): return simplify(binomial(2*n, k)*gamma(k)*hypergeometric([(1-k)/2, -k/2], [1-k, 1/2-n], -2)/2)
flatten([[A167986(n,k) for k in range(3,2*n+1)] for n in range(2,11)]) # G. C. Greubel, Jan 17 2023

T(n,k) = Sum_{j=0..floor(k/2)} (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!/2. - Andrew Howroyd, May 09 2017

A167981 Number of 2n-cycles on the graph of the tesseract, 2 <= n <= 8.

Original entry on oeis.org

24, 128, 696, 2112, 5024, 5736, 1344

Offset: 2

Andrew Weimholt, Nov 16 2009

Row n=4 of the triangle in A085452
The graph of any n-cube (n>1) contains only even length cycles.
The tesseract is the 4 dimensional cube, and is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the tesseract is {4,3,3}.

			a(2) = 24 because there are 24 4-cycles on the graph of the tesseract.
The cycle polynomial is  24*x^4 + 128*x^6 + 696*x^8 + 2112*x^10 + 5024*x^12 + 5376*x^14 + 1344*x^16.

A. Weimholt, Tesseract Foldout
Eric Weisstein's World of Mathematics, Cycle Polynomial
Eric Weisstein's World of Mathematics, Tesseract Graph

Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

A167982 Number of n-cycles on the graph of the regular 16-cell, 3 <= n <= 8.

Original entry on oeis.org

32, 102, 288, 640, 960, 744

Offset: 3

Andrew Weimholt, Nov 16 2009

Row n=3 of the triangle in A167986

The 16-cell is the dual polytope of the tesseract, and is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the 16-cell is {3,3,4}.

			a(3) = 32, because there are 32 3-cycles on the graph of the 16-cell.
Cycle polynomial is 32*x^3 + 102*x^4 + 288*x^5 + 640*x^6 + 960*x^7 + 744*x^8.

A. Weimholt, 16-cell net
Eric Weisstein's World of Mathematics, 16-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

A167983 Number of n-cycles on the graph of the regular 24-cell, 3 <= n <= 24.

Original entry on oeis.org

96, 360, 1440, 7120, 37728, 196488, 974592, 4536000, 19934208, 82689264, 322437312, 1171745280, 3924079104, 11964375936, 32761139328, 79244294016, 165800420352, 291640320576, 413774810112, 443415854592, 318534709248, 114869295744

Offset: 3

Andrew Weimholt, Nov 16 2009

The 24-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol of the 24-cell is {3,4,3}.

			a(3) = 96, because there are 96 3-cycles on the graph of the 24-cell.
Cycle polynomial is 96*x^3 + 360*x^4 + 1440*x^5 + 7120*x^6 + 37728*x^7 + 196488*x^8 + 974592*x^9 + 4536000*x^10 + 19934208*x^11 + 82689264*x^12 + 322437312*x^13 + 1171745280*x^14 + 3924079104*x^15 + 11964375936*x^16 + 32761139328*x^17 + 79244294016*x^18 + 165800420352*x^19 + 291640320576*x^20 + 413774810112*x^21 + 443415854592*x^22 + 318534709248*x^23 + 114869295744*x^24.

Max Alekseyev, PARI/GP scripts for various math problems
Max A. Alekseyev, Gérard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016.
A. Weimholt, 24-cell net
Eric Weisstein's World of Mathematics, 24-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

a(16)-a(24) and "full" keyword from Max Alekseyev, Nov 18 2009

A167984 Number of n-cycles on the graph of the regular 120-cell, 3 <= n <= 600.

Original entry on oeis.org

0, 0, 720, 0, 0, 3600, 2400, 4320, 28800, 35400, 64800, 284400, 540000, 1139400, 3708000, 8557200, 19677600, 55725120, 140359200, 346456800, 935942400, 2442469200, 6282571680

Offset: 3

Andrew Weimholt, Nov 16 2009

The 120-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol of the 120-cell is {5,3,3}.

			a(5) = 720, because there are 720 5-cycles on the graph of the 120-cell.
Cycle polynomial is 720*x^5 + 3600*x^8 + 2400*x^9 + 4320*x^10 + 28800*x^11 + 35400*x^12 + 64800*x^13 +  284400*x^14 + 540000*x^15 + 1139400*x^16 + 3708000*x^17 + 8557200*x^18 + 19677600*x^19 + 55725120*x^20 + 140359200*x^21 + 346456800*x^22 + 935942400*x^23 + ...

A. Weimholt, 120-cell net
Eric Weisstein's World of Mathematics, 120-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).
Cf. A108997 (number of vertices n-steps from a given vertex on graph of 120-cell).

a(24) from Eric W. Weisstein, Feb 21 2014

a(25) from Eric W. Weisstein, Mar 11 2014

A167985 Number of n-cycles on the graph of the regular 600-cell, 3 <= n <= 120.

Original entry on oeis.org

1200, 5400, 29520, 187200, 1310400, 9813600, 77193600, 630538632, 5307656400

Offset: 3

Andrew Weimholt, Nov 16 2009

The 600-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the 600-cell is {3,3,5}.

			a(3) = 1200, because there are 1200 3-cycles on the graph of the 600-cell.
Cycle polynomial is 1200*x^3 + 5400*x^4 + 29520*x^5 + 187200*x^6 + 1310400*x^7 + 9813600*x^8 + 77193600*x^9 + 630538632*x^10 + ...

A. Weimholt, 600-cell net
Eric Weisstein's World of Mathematics, 600-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).
Cf. A118785 (number of vertices n-steps from a given vertex on graph of the 600-cell).

a(11) from Eric W. Weisstein, Feb 09 2014

A116723 We have one bead labeled i for every i=1, 2, ...; a(n) = number of necklaces that can be made using any subset of the first n beads.

Original entry on oeis.org

1, 2, 4, 8, 18, 53, 219, 1201, 8055, 62860, 556070, 5488126, 59740688, 710771367, 9174170117, 127661752527, 1904975488573, 30341995265190, 513771331467544, 9215499383109764, 174548332364311774, 3481204991988351785, 72920994844093191807, 1600596371590399672061

Offset: 0

Rodney Stephenson (rod.stephenson(AT)gmail.com), Mar 19 2008

nonn

Turning the necklace over doesn't count as a different necklace.

For k beads chosen from n distinct ones, we can have n!/(n-k)! possible permutations, then eliminate the cyclic permutations and reversals (2k). The case k=2 is special in that a cyclic permutation is equivalent to a reversal.

			For example for n=4 we have {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, {1,2,4,3}, {1,3,2,4}.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Qu Cao, Jin Dong, Song He, and Fan Zhu, On one-loop amplitudes in gauge theories, arXiv:2412.19629 [hep-th], 2024. See p. 4.
Index entries for sequences related to necklaces

Row sums of A144151. - Alois P. Heinz, Jun 01 2009

a:= proc(n) option remember; `if`(n<4, 2^n, `if`(n=4, 18,
      ((n^3-4*n^2+n)*a(n-1) -(2*n-2)*(n^2-4*n+2)*a(n-2)
       +n*(n-2)*(n-3)*a(n-3)) / ((n-1)*(n-4))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 22 2016

a[n_] := 1 + n + n(n-1)/2 + Sum[n!/(2k(n-k)!), {k, 3, n}];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 09 2020 *)

a(n) = 1 + n + n*(n-1)/2 + sum(k=3, n, n!/(2*k*(n-k)!)); \\ Michel Marcus, Nov 09 2020

a(n) = 1 + n + n(n-1)/2 + Sum_{k=3..n} n!/(2k(n-k)!).

More terms from Washington Bomfim, Aug 29 2008

A284947 Irregular triangle read by rows: coefficients of the cycle polynomial of the n-complete graph K_n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 4, 3, 0, 0, 0, 10, 15, 12, 0, 0, 0, 20, 45, 72, 60, 0, 0, 0, 35, 105, 252, 420, 360, 0, 0, 0, 56, 210, 672, 1680, 2880, 2520, 0, 0, 0, 84, 378, 1512, 5040, 12960, 22680, 20160, 0, 0, 0, 120, 630, 3024, 12600, 43200, 113400, 201600, 181440

Offset: 3

Eric W. Weisstein, Apr 06 2017

			1: 0
2: 0
3: x^3
4: x^3 (4 + 3 x)
5: x^3 (10 + 15 x + 12 x^2)
6: x^3 (20 + 45 x + 72 x^2 + 60 x^3)
giving
1 3-cycle in K_3
4 3-cycles and 3 4-cycles in K_4
From _Peter Luschny_, Oct 22 2017: (Start)
Prepending six zeros leads to the regular triangle:
[0] 0
[1] 0, 0
[2] 0, 0, 0
[3] 0, 0, 0,  1
[4] 0, 0, 0,  4,   3
[5] 0, 0, 0, 10,  15,   12
[6] 0, 0, 0, 20,  45,   72,   60
[7] 0, 0, 0, 35, 105,  252,  420,   360
[8] 0, 0, 0, 56, 210,  672, 1680,  2880,  2520
[9] 0, 0, 0, 84, 378, 1512, 5040, 12960, 22680, 20160
(End)

Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A002807 (row sums of a(n)).

Cf. A144151 (generalization to include 1- and 2-"cycles").

A284947row := n -> seq(`if`(k<3, 0, pochhammer(3,k-3)*binomial(n,k)), k=0..n):
seq(A284947row(n), n=3..10); # Peter Luschny, Oct 22 2017

CoefficientList[Table[-(n*x*(2 - x + n*x - 2*HypergeometricPFQ[{1, 1, 1 - n}, {2}, -x]))/4, {n, 10}], x] // Flatten

T(n, k) = binomial(n, k)*Pochhammer(3, k-3) if k >= 3 else 0. - Peter Luschny, Oct 22 2017

A296546 Triangle read by rows T(n,k): number of undirected cycles of length k in the complete tripartite graph K_{n,n,n} (n = 1...; k = 3..3n).

Original entry on oeis.org

1, 8, 15, 24, 16, 27, 108, 324, 774, 1620, 2268, 1584, 64, 396, 1728, 7200, 27648, 87480, 232704, 476928, 663552, 463104, 125, 1050, 6000, 35800, 198000, 977400, 4392000, 17068320, 56376000, 151632000, 311040000, 430272000, 299289600

Offset: 1

Eric W. Weisstein, Dec 15 2017

			Written as cycle polynomials:
  x^3
  8 x^3 + 15 x^4 + 24 x^5 + 16 x^6
  27 x^3 + 108 x^4 + 324 x^5 + 774 x^6 + 1620 x^7 + 2268 x^8 + 1584 x^9
  64 x^3 + 396 x^4 + 1728 x^5 + 7200 x^6 + 27648 x^7 + 87480 x^8 + 232704 x^9 + 476928 x^10 + 663552 x^11 + 463104 x^12
giving the array
  1
  8, 15, 24, 16
  27, 108, 324, 774, 1620, 2268, 1584
  64, 396, 1728, 7200, 27648, 87480, 232704, 476928, 663552, 463104

Eric W. Weisstein, Table of n, a(n) for n = 1..117 (rows n = 1..9, flattened).
Eric Weisstein's World of Mathematics, Cycle Polynomial.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph.

Cf. A234616 (number of undirected cycles in K_{n,n,n}).
Cf. A144151 (cycle polynomial coefficients of complete graph K_n).
Cf. A291909 (cycle polynomial coefficients of complete bipartite graph K_{n,n}).

Table[Tally[Length /@ FindCycle[CompleteGraph[{n, n, n}], Infinity, All]][[All, 2]], {n, 4}] // Flatten

Row sums of T(n,k) give A234616(n).

cp's OEIS Frontend

A167986 Triangle T(n,k) = Number of k-cycles on the graph of an n-orthoplex. n>=2, k>=3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A167981 Number of 2n-cycles on the graph of the tesseract, 2 <= n <= 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A167982 Number of n-cycles on the graph of the regular 16-cell, 3 <= n <= 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A167983 Number of n-cycles on the graph of the regular 24-cell, 3 <= n <= 24.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A167984 Number of n-cycles on the graph of the regular 120-cell, 3 <= n <= 600.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A167985 Number of n-cycles on the graph of the regular 600-cell, 3 <= n <= 120.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A116723 We have one bead labeled i for every i=1, 2, ...; a(n) = number of necklaces that can be made using any subset of the first n beads.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A284947 Irregular triangle read by rows: coefficients of the cycle polynomial of the n-complete graph K_n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs