A214313 -id:A214313 - cp's OEIS frontend

A214311 a(n) is the number of representative five-color bracelets (necklaces with turning over allowed) with n beads, for n >= 5.

12, 30, 150, 633, 3260, 16212, 66810, 298495, 1410402, 6403842, 31103899, 135342046, 633228696, 2936824916, 13676037486, 65355191817, 298065986582, 1398226666434, 6585151203697, 30958838054304, 148994847644780

Offset: 5

Wolfdieter Lang, Aug 08 2012

nonn

This is the fifth column (m=5) of triangle A213940.
The relevant p(n,5)= A008284(n,5) representative color multinomials have exponents (signatures) from the five-part partitions of n, written with nonincreasing parts. E.g., n=7: [3,1,1,1,1] and [2,2,1,1,1] (p(7,5)=2). The corresponding representative bracelets have the five-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5] and c[1]^2*c[2]^2*c[3]*c[4]*c[5].
Number of n-length bracelets w over a 5-ary alphabet {a1,a2,...,a5} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a5) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226884). The number of 5 color bracelets up to permutations of colors is given by A056360. - Andrew Howroyd, Sep 26 2017

			a(5) = A213940(5,5) = A213939(5,7) = 12 from the representative bracelets (with colors j for c[j], j=1,...,5) 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235 and 14325, all taken cyclically.

Andrew Howroyd, Table of n, a(n) for n = 5..200

Cf. A213939, A213940, A214309 (m=4 case), A214313 (m=5, all bracelets).

Cf. A056360, A226884.

a(n) = A213940(n,5), n >= 5.

a(n) = sum(A213939(n,k),k= b(n,5)..b(n,6)-1), n>=6, with b(n,m) = A214314(n,m) the position where the first m part partition of n appears in the Abramowitz-Stegun ordering of partitions (see A036036 for the reference and a historical comment). a(5) = A213939(5,b(5,5)) = A213939(5,7) = 12.

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

cp's OEIS Frontend

A214311 a(n) is the number of representative five-color bracelets (necklaces with turning over allowed) with n beads, for n >= 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula