A056342 -id:A056342 - cp's OEIS frontend

A273891 Triangle read by rows: T(n,k) is the number of n-bead bracelets with exactly k different colored beads.

1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 18, 24, 12, 1, 11, 56, 136, 150, 60, 1, 16, 147, 612, 1200, 1080, 360, 1, 28, 411, 2619, 7905, 11970, 8820, 2520, 1, 44, 1084, 10480, 46400, 105840, 129360, 80640, 20160, 1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440

Offset: 1

Marko Riedel, Jun 02 2016

For bracelets, chiral pairs are counted as one.

			Triangle begins with T(1,1):
1;
1,  1;
1,  2,    1;
1,  4,    6,     3;
1,  6,   18,    24,     12;
1, 11,   56,   136,    150,     60;
1, 16,  147,   612,   1200,   1080,     360;
1, 28,  411,  2619,   7905,  11970,    8820,    2520;
1, 44, 1084, 10480,  46400, 105840,  129360,   80640,  20160;
1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440;
For T(4,2)=4, the arrangements are AAAB, AABB, ABAB, and ABBB, all achiral.
For T(4,4)=3, the arrangements are ABCD, ABDC, and ACBD, whose chiral partners are ADCB, ACDB, and ADBC respectively. - _Robert A. Russell_, Sep 26 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Marko Riedel, Maple code for A087854 and A273891.

Columns 1-6: A057427, A056342, A056343, A056344, A056345, A056346.
Row sums give A019537.
Cf. A087854 (oriented), A305540 (achiral), A305541 (chiral).

(* t = A081720 *) t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n+1)/2))/(2*n)]); T[n_, k_] := Sum[(-1)^i * Binomial[k, i]*t[n, k-i], {i, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 07 2017, after Andrew Howroyd *)
Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,10}, {k,1,n}] // Flatten (* Robert A. Russell, Sep 26 2018 *)

T(n,k) = Sum_{i=0..k-1} (-1)^i * binomial(k,i) * A081720(n,k-i). - Andrew Howroyd, Mar 25 2017
From Robert A. Russell, Sep 26 2018: (Start)
T(n,k) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where S2 is the Stirling subset number A008277.
G.f. for column k>1: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j*x^d).
T(n,k) = (A087854(n,k) + A305540(n,k)) / 2 = A087854(n,k) - A305541(n,k) = A305541(n,k) + A305540(n,k).
(End)

A091696 Number of classes of compositions of n equivalent under reflection or cycling.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 17, 29, 45, 77, 125, 223, 379, 686, 1223, 2249, 4111, 7684, 14309, 27011, 50963, 96908, 184409, 352697, 675187, 1296857, 2493725, 4806077, 9272779, 17920859, 34669601, 67159049, 130216123, 252745367, 490984487, 954637557, 1857545299

Offset: 1

Neil Fernandez, Jan 29 2004

nonn

Equivalently, the number of radial cutting patterns of a (maximally symmetric) circular cake such that all resulting pieces are a multiple of 1/n of the whole. - Peter Munn, Oct 27 2020

			7 has 15 partitions and 64 compositions. Compositions can be mapped to other compositions by reflection, cycling, or both, e.g., {1,2,4} -> {4,2,1} (reflection), {2,4,1} (cycling), or {1,4,2} (both); this defines the equivalence relation used. The number of equivalence classes so defined is 2 greater than the number of partitions because only {3,1,2,1} and {2,1,2,1,1} (and their equivalents) cannot be mapped to the conventionally stated forms of partitions (here, {3,2,1,1} and {2,2,1,1,1} respectively). So a(7) = 15 + 2 = 17.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000029, A056342.

DIK transform of A057427.

with(numtheory):
a:= n-> add(phi(d)*2^(n/d)/(2*n), d=divisors(n))
        +`if`(irem(n, 2)=0, 2^(n/2-1) +2^(n/2-2), 2^((n-1)/2)) -1:
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 20 2012

Needs["Combinatorica`"]
nn=40;Apply[Plus,Table[CoefficientList[Series[CycleIndex[DihedralGroup[n], s]/.Table[s[i]->x^i/(1-x^i),{i,1,nn}],{x,0,nn}],x],{n,1,nn}]]  (* Geoffrey Critzer, Oct 18 2012 *)
mx:=50;CoefficientList[Series[(Sum[(EulerPhi[n] Log[2+1/(-1+x^n)])/n,{n,1,mx}]+(1-1x^2+ x^3)/((x-1) (1-2 x^2)))/(-2),{x,0,mx}],x] (* Herbert Kociemba, Dec 04 2016 *)
a[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#) & ]/(2*n) - 1; Array[a, 37] (* Jean-François Alcover, Nov 05 2017 *)

a(n) = A000029(n) - 1.
a(n) = A056342(n) + 1.
G.f.: ( Sum_{n>=1} phi(n)*log(2+1/(-1+x^n))/n + (1-1x^2+x^3)/((x-1)*(1-2*x^2)) )/(-2). - Herbert Kociemba, Dec 04 2016

More terms from Sean A. Irvine, Feb 09 2012

A214308 a(n) is the number of all two colored bracelets (necklaces with turning over allowed) with n beads with the two colors from a repertoire of n distinct colors, for n>=2.

Original entry on oeis.org

1, 6, 24, 60, 165, 336, 784, 1584, 3420, 6820, 14652, 29484, 62335, 128310, 269760, 558960, 1175499, 2446668, 5131900, 10702020, 22385517, 46655224, 97344096, 202555800, 421478200, 875297124, 1816696728, 3764747868, 7795573230, 16121364000, 33310887808

Offset: 2

Wolfdieter Lang, Jul 31 2012

nonn

This is the second column (m=2) of triangle A214306.
Each 2 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2]], with p[1] >= p[2] >= 1, there are A213941(n,k)= A035206(n,k)*A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,2)= A008284(n,2) = floor(n/2) partitions of n with 2 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
Compare this sequence with A000029 where also single colored bracelets are included, and the color repertoire is only [c[1], c[2]] for all n.

			a(5) = A213941(5,2) + A213941(5,3) = 20 + 40 = 60 from the bracelet (with colors j for c[j], j=1,2,..,5) cyclic(11112) which represents a class of order A035206(5,2) = 20 (if all 5 colors are used), cyclic(11122) and cyclic(11212) each representing also a color class of 20 members each, summing to 60 bracelets  with five beads and five colors available for the two color signatures [4,1] and [3,2].

Andrew Howroyd, Table of n, a(n) for n = 2..100

Cf. A213941, A214306, A213942 (m=2, representative bracelets), A214310 (m=3).

a(n) = A214306(n,2), n >= 2.
a(n) = sum(A213941(n,k),k=2..A008284(n,2)+1), n>=2, with A008284(n,2) = floor(n/2).
a(n) = binomial(n,2) * A056342(n). - Andrew Howroyd, Mar 25 2017

a(25)-a(32) from Andrew Howroyd, Mar 25 2017

A103442 Row sums of A103441. Number of two-colored bracelets of n beads with different sets of distances among the white beads.

Original entry on oeis.org

1, 2, 4, 6, 11, 16, 27, 44, 73, 124, 199, 372, 613, 1142, 1874, 3926, 6209, 13660, 21539, 46240, 80142

Offset: 2

Wouter Meeussen, Feb 06 2005

Cf. A103441, A056342.

(* see A103441 *) Table[Length[Union[(dist[f[ #1], n]&)/@Flatten[Table[ListNecklaces[n, Join[1+0*Range[i], 0*Range[n-i]], Dihedral], {i, 1, n-1}], 1]]], {n, 2, 16}]

a(21) & a(22) from Robert G. Wilson v, Aug 09 2010

A375617 Numbers of facially complete 2-connected planar embeddings.

Original entry on oeis.org

0, 0, 1, 3, 6, 15, 32, 94, 295, 1169, 4870, 22110, 102490, 489479, 2370856, 11655722, 57918613, 290697549, 1471349079, 7504192109, 38532719288, 199076246027, 1034236802988, 5400337234593, 28329240686563, 149244907924935, 789351357094770, 4190055030317638

Offset: 1

Eric W. Weisstein, Aug 21 2024

nonn

Eric W. Weisstein, Table of n, a(n) for n = 1..100
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.
Eric Weisstein's World of Mathematics, Facially Complete Planar Embedding.

prism[n_] := Floor[((n - 3)^2 + 6)/12]
tetrahedral[n_] := prism[n - 1]
bipartite[n_] := prism[n - 2]
wheel[n_] := (Mod[n - 1, 2] + 3) 2^Quotient[n - 1, 2]/4 + DivisorSum[n - 1, EulerPhi[#] 2^((n - 1)/#) &]/(2 (n - 1)) - 3
cycle[n_] := Module[{f, F, x},
  f[x_, m_] := x + Sum[(Binomial[s - 2, r - 1] Binomial[r + s - 1, s] x^s)/r, {r, m}, {s, 2, m}];
  F[x_, m_] := Series[((3 x^2 - 2 x f[x, m] + f[x, m]^2) - (2 + 2 x + 7 x^2 - 4 x f[x, m] + 2 f[x, m]^2) f[x^2, m] + 2 f[x^2, m]^2)/(4 (2 f[x^2, m] - 1)) + Sum[If[Mod[k, d] == 0, (EulerPhi[d] f[x^d, m]^(k/d))/k, 0], {k, 3, m}, {d, k}]/2, {x, 0, m}];
  CoefficientList[F[x, n], x][[-1]]]
a[1] = a[2] = 0;
a[n_] := prism[n] + tetrahedral[n] + bipartite[n] + wheel[n] + cycle[n]
Table[a[n], {n, 20}]

a(n) = A001399(n - 6) + A001399(n - 7) + A001399(n - 8) + (A056342(n - 1) - 1) + A001004(n).

A056348 Number of primitive (period n) bracelets using exactly two different colored beads.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 16, 24, 42, 69, 124, 208, 378, 668, 1214, 2220, 4110, 7630, 14308, 26931, 50944, 96782, 184408, 352450, 675180, 1296477, 2493680, 4805388, 9272778, 17919558, 34669600

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet. Identical to A001371 for n>1.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A001037.

Sum mu(d)*A056342(n/d) where d divides n.

cp's OEIS Frontend

A273891 Triangle read by rows: T(n,k) is the number of n-bead bracelets with exactly k different colored beads.

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A091696 Number of classes of compositions of n equivalent under reflection or cycling.

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A214308 a(n) is the number of all two colored bracelets (necklaces with turning over allowed) with n beads with the two colors from a repertoire of n distinct colors, for n>=2.

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A103442 Row sums of A103441. Number of two-colored bracelets of n beads with different sets of distances among the white beads.

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A375617 Numbers of facially complete 2-connected planar embeddings.

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A056348 Number of primitive (period n) bracelets using exactly two different colored beads.

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