A056344 -id:A056344 - 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)

A056490 Number of periodic palindromes using exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 12, 24, 132, 240, 900, 1560, 4980, 8400, 24612, 40824, 113652, 186480, 502500, 818520, 2158260, 3498000, 9087012, 14676024, 37728372, 60780720, 155091300, 249401880, 632972340, 1016542800, 2569858212, 4123173624, 10393634292, 16664094960

Offset: 1

Marks R. Nester

			For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
For n=6, the 12 arrangements are ABACDC, ABADCD, ACABDB, ACADBD, ADABCB, ADACBC, ABCDCB, ABDCDB, ACBDBC, ACDBDC, ADBCBD, and ADCBCD.

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]

Muniru A Asiru, Table of n, a(n) for n = 1..700
Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-26,26,24,-24).

Cf. A056284, A056344, A056455.

Column 4 of A305540.

a:=[0,0,0,0,0,12,24];; for n in [8..35] do a[n]:=a[n-1]+9*a[n-2]-9*a[n-3]-26*a[n-4]+26*a[n-5]+24*a[n-6]-24*a[n-7]; od; a; # Muniru A Asiru, Sep 26 2018

m:=50; R:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0] cat Coefficients(R!(12*x^6*(1+x)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)*(1-3*x^2)))); // G. C. Greubel, Oct 13 2018

a:=n->(factorial(4)/2)*(Stirling2(floor((n+1)/2),4)+Stirling2(ceil((n+1)/2),4)): seq(a(n),n=1..35); # Muniru A Asiru, Sep 26 2018

k = 4; Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] +
StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 40}] (* Robert A. Russell, Jun 05 2018 *)
LinearRecurrence[{1,9,-9,-26,26,24,-24}, {0,0,0,0,0,12,24}, 40] (* Robert A. Russell, Sep 29 2018 *)

a(n) = my(k=4); (k!/2)*(stirling(floor((n+1)/2), k, 2) + stirling(ceil((n+1)/2), k, 2)); \\ Michel Marcus, Jun 05 2018

a(n) = 2*A056344(n) - A056284(n).
G.f.: 12*x^6*(1+x)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)*(1-3*x^2)). - Colin Barker, May 06 2012
a(n) = (k!/2)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), with k=4 different colors used and where S2(n,k) is the Stirling subset number A008277. - Robert A. Russell, Jun 05 2018

A214312 a(n) is the number of all four-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 4.

Original entry on oeis.org

3, 120, 2040, 21420, 183330, 1320480, 8691480, 52727400, 303958710, 1674472800, 8928735816, 46280581620, 234611247780, 1166708558400, 5710351190400, 27565250985360, 131495088522060, 620771489730000, 2903870526350640, 13473567673441260, 62061657617625204, 283995655732351200

Offset: 4

Wolfdieter Lang, Jul 31 2012

nonn

This is the fourth column (m=4) of triangle A214306.
Each 4 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], p[3], p[4]], with p[1] >= p[2]  >= p[3] >= p[4] >= 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,4)= A008284(n,4)  partitions of n with 4 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
Compare this with A032275 where also bracelets with less than four colors are included, and the color repertoire is only [c[1], c[2], c[3], c[4]] for all n.

			a(5) = A213941(5,6) = 120 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11234, 11243, 11324, 12134, 13124 and 14123, all six taken cyclically, each representing a class of order A035206(5,6) = 20 (if all 5 colors are used). For example, cyclic(11342) becomes equivalent to cyclic(11243) by turning over or reflection. The multiplicity 20 depends only on the color signature.

Andrew Howroyd, Table of n, a(n) for n = 4..100

Cf. A213941, A214306, A214309 (m=4, representative bracelets), A214313 (m=5).

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)]);
a56344[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
a[n_] := Binomial[n, 4]*a56344[n, 4];
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

a(n) = A214306(n,4), n >= 4.
a(n) = sum(A213941(n,k),k = A214314(n,4) .. (A214314(n,4) - 1 + A008284(n,4))), n >= 4.
a(n) = binomial(n,4) * A056344(n). - Andrew Howroyd, Mar 25 2017

A326789 Number of n-bead asymmetric bracelets with exactly 4 different colored beads.

Original entry on oeis.org

0, 0, 0, 3, 24, 124, 588, 2484, 10240, 40464, 158220, 608951, 2333520, 8894616, 33864340, 128791140, 490027200, 1865614976, 7110959340, 27138170397, 103717719412, 396965536224, 1521562700988, 5840509149020, 22450188684264, 86412086034120, 333035003533660

Offset: 1

Andrew Howroyd, Sep 12 2019

nonn

Asymmetric bracelets are those primitive (period n) bracelets that when turned over are different from themselves.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column k=4 of A309651.

Cf. A032239, A032240, A032241, A056344, A056350, A305543, A326660.

a(n) = A032241(n) - 4*A032240(n) + 6*A032239(n) for n >= 3.

Moebius transform of A305543.

A056350 Number of primitive (period n) bracelets using exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 3, 24, 136, 612, 2616, 10480, 41364, 159780, 613919, 2341920, 8919204, 33905164, 128904660, 490213680, 1866117224, 7111777860, 27140327757, 103721217388, 396974621676, 1521577377012, 5840546872280

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet.

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. A056289.

Sum mu(d)*A056344(n/d) where d|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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A056490 Number of periodic palindromes using exactly four different symbols.

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A214312 a(n) is the number of all four-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 4.

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A326789 Number of n-bead asymmetric bracelets with exactly 4 different colored beads.

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A056350 Number of primitive (period n) bracelets using exactly four different colored beads.

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