A214311 -id:A214311 - cp's OEIS frontend

A213940 Triangle with entry a(n,m) giving the number of bracelets of n beads (dihedral D_n symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.

1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 1, 3, 6, 6, 12, 1, 7, 20, 26, 30, 60, 1, 8, 40, 93, 150, 180, 360, 1, 18, 106, 424, 633, 1050, 1260, 2520, 1, 22, 304, 1180, 3260, 5040, 8400, 10080, 20160, 1, 46, 731, 4844, 16212, 29244, 45360, 75600, 90720, 181440

Offset: 1

Wolfdieter Lang, Jul 20 2012

This triangle is obtained from the partition array A213939 by summing in row n, for n>=1, all entries related to partitions of n with the same number of parts m.
a(n,m) is the number of bracelets of n beads (dihedral D_n symmetry) corresponding to the representative color multinomials obtained from all partitions of n with m parts by 'exponentiation', hence only m from the available n colors are present. As a representative multinomial of each of the p(n,m)=A008284(n,m) such m-color classes we take the one where the considered m part partition of n, [p[1],...,p[m]], written in nonincreasing order, is distributed as exponents on the color indices like c[1]^p[1]*...*c[m]^p[m]. That is only the first m colors from the n available ones are involved.
See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions.
The row sums of this triangle coincide with the ones of array A213939, and they are given by A213943.
Number of n-length bracelets w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226874). - Andrew Howroyd, Sep 26 2017

			n\m  1  2   3    4     5     6     7     8     9     10 ...
1    1
2    1  1
3    1  1   1
4    1  3   2    3
5    1  3   6    6    12
6    1  7  20   26    30    60
7    1  8  40   93   150   180   360
8    1 18 106  424   633  1050  1260  2520
9    1 22 304 1180  3260  5040  8400 10080 20160
10   1 46 731 4844 16212 29244 45360 75600 90720 181440
...
a(5,3) = 2 + 4 = 6, from A213939(5,4) + A213939(5,5), because k(5,3,1) = 4 and p(5,3) = 2.
a(2,1) = 1 because the partition [2] of n=2 with part number m=1 corresponds to the representative color multinomial (here monomial) c[1]^2 = c[1]*c[1], and there is one such representative bracelet. There is another bracelet color monomial in this class of n=2 colors where only m=1 color is active: c[2]*c[2]. See the triangle entry A213941(2,1)=2. The same holds for the necklace case.
a(3,1) = 1 from the color monomial representative c[1]^3. This class has 2 other members: c[2]^3 and c[3]^3. See A213941(3,1)=3. The same holds for the necklace case.
Like in the necklace case one has in general a(n,1)=1 and A213941(n,1) = n from the partition [n] providing the color signature and a representative c[1]^n.
a(3,2) = 1 from the representative color multinomial c[1]^2*c[2] (from the m=2 partition [2,1] of n=3) leading to just one representative bracelet (and necklace) cyclic(112) (when one uses j for color c[j]). The whole class consists of A213941(3,2)=6 bracelets (or necklaces): cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332).
a(3,3) = 1. The representative color multinomial is c[1]*c[2]*c[3] (from the m=3 partition [1,1,1]). There is only one bracelet cyclic(1,2,3) which constitutes already the whole class (A213941(3,3)=1). The necklace cyclic(1,3,2) becomes equivalent under D_3.
a(4,2) = 3 from two representative color multinomials c[1]^3*c[2] and c[1]^2*c[2]^2 (from the two m=2 partitions of n=4: [3,1] and [2,2]). The first one has one representative bracelet, namely cyclic(1112), the second one leads to the two representative bracelets: cyclic(1122) and cyclic(1212). Together these are the 3 bracelets counted by a(4,2). The first color class c[.]^3*c[.] consists of 4*3=12 bracelets, when all 4 colors are used. The second one consists of 2*6=12 bracelets. Together they sum up to the 24 bracelets counted by A213941(4,2). In this example the necklace case does not differ from the bracelet one.

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=2..5 are A213942, A214307, A214309, A214311.
Cf. A008284, A213939, A213941, A213943 (row sums), A226874, A152176.
Cf. A213934 (cyclic symmetry).

Cyc(v)={my(s=vecsum(v)); sumdiv(gcd(v), d, eulerphi(d)*(s/d)!/prod(i=1, #v, (v[i]/d)!))/s}
CPal(v)={my(odds=#select(t->t%2,v), s=vecsum(v));  if(odds>2, 0, ((s-odds)/2)!/prod(i=1, #v, (v[i]\2)!))}
T(n,k)={my(t=0); forpart(p=n, t+=Cyc(Vec(p))+CPal(Vec(p)), [1,n], [k,k]); t/2}
for(n=1, 10, for(k=1,n, print1(T(n,k), ", ")); print); \\ Andrew Howroyd, Sep 26 2017

\\ faster version; here U is A226874 as vector of polynomials.
U(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
T(n)={my(t=U(n)); vector(n, n, vector(n, k, ((1/n)*sumdiv(n, d, eulerphi(n/d) * polcoeff(t[d+1], k)) + if(n%2, sum(d=0, (n-1)/2, binomial((n-1)/2, d)*polcoeff(t[d+1], (k-1))), polcoeff(t[n/2+1], k) + sum(d=0, n/2-1, binomial(n/2-1, d)*(2^d + if(d%2, 0, binomial(d, d/2)))*polcoeff(t[n/2-d], k-2))/2))/2))}
{ my(t=T(10)); for(n=1, #t, print(t[n])) } \\ Andrew Howroyd, Dec 22 2017

a(n,m) = Sum_{j=1..p(n,m)}A213939(n,k(n,m,1)+j-1), with k(n,m,1) the position where in the list of partitions of n in A-St order the first with m parts appears, and p(n,m) is the number of partitions of n with m parts shown in the array A008284. E.g., n=5, m=3: k(5,3,1)=4, p(5,3)=2.

A214309 a(n) is the number of representative four-color bracelets (necklaces with turning over allowed) with n beads, for n >= 4.

Original entry on oeis.org

3, 6, 26, 93, 424, 1180, 4844, 16165, 66953, 216804, 852822, 2949804, 12119134, 40886724, 160826008, 572457489, 2331396595, 8104270828, 32043699894, 115995102806, 471268872328, 1674576998468, 6641876380417, 24364816845446, 98894256728960, 357006263815751

Offset: 4

Wolfdieter Lang, Jul 31 2012

nonn

This is the fourth column (m=4) of triangle A213940.
The relevant p(n,4)= A008284(n,4) representative color multinomials have exponents (signatures) from the 4 part partitions of n, written with nonincreasing parts. E.g., n=6: [3,1,1,1] and [2,2,1,1] (p(6,4)=2). The corresponding representative bracelets have the four-color multinomials c[1]^3*c[2]*c[3]*c[4] and c[1]^2*c[2]^2*c[3]*c[4].
Compare this with A032275 where also bracelets with less than four colors are included, and not only representatives are counted.
Number of n-length bracelets w over a 4-ary alphabet {a1,a2,...,a4} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a4) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226883). The number of 4 color bracelets up to permutations of colors is given by A056359. - Andrew Howroyd, Sep 26 2017

			a(4) = A213939(4,5) = 3 from the representative bracelets (with colors  j for c[j], j=1, 2, ..., 4) 1234, 1342 and 1423, all taken cyclically. The necklace cyclic(1324), for example, becomes equivalent to cyclic(1423) under the dihedral D_4 turning over (or reflection) operation.
a(6) = A213939(6, 8) = A213939(6, 9) =  10 + 16 = 26. See the comment above for the representative color multinomials for each case.

Andrew Howroyd, Table of n, a(n) for n = 4..200

Cf. A213939, A213940, A214311 (m=5), A214312 (m=4, all bracelets).

Cf. A056359, A226883.

Cyc(v)={my(g=fold(gcd,v), s=vecsum(v)); sumdiv(g, d, eulerphi(d)*(s/d)!/prod(i=1, #v, (v[i]/d)!))/s}
CPal(v)={my(odds=#select(t->t%2,v), s=vecsum(v));  if(odds>2, 0, ((s-odds)/2)!/prod(i=1, #v, (v[i]\2)!))}
a(n)={my(t=0); forpart(p=n, t+=Cyc(Vec(p))+CPal(Vec(p)), [1,n], [4,4]); t/2} \\ Andrew Howroyd, Sep 26 2017

a(n) = A213940(n,4), n >= 4.

a(n) = sum(A213939(n,k),k=(2+floor(n/2) + p(n,3))..(p(n,4)+1+floor(n/2)+p(n,3))), n>=4, with p(n,m) = A008284(n,m) the number of partitions of n with m parts.

Terms a(26) and beyond from Andrew Howroyd, Sep 26 2017

A056360 Number of bracelet structures using exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 16, 85, 434, 2271, 11530, 58400, 290689, 1436685, 7036418, 34286464, 166316979, 804557406, 3884248150, 18731033958, 90269841924, 434955114981, 2096028083116, 10104206901987, 48733744753173, 235196202817401, 1135892957109815, 5490007141743186

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

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]

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

Column 5 of A152176.

Cf. A056298, A056354, A056355, A214311.

a(n) = A056355(n) - A056354(n).

Terms a(25) and beyond from Andrew Howroyd, Oct 24 2019

A214313 a(n) is the number of all five-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 5.

Original entry on oeis.org

12, 900, 25200, 442680, 5846400, 64420272, 622175400, 5466166200, 44611306740, 343916472900, 2531921456064, 17956666859040, 123458676825120, 827056125453600, 5419508203393200, 34847210197637424, 220424306985639540, 1374479672119161300, 8463477229726134000, 51536194734146965920, 310706598354410079360

Offset: 5

Wolfdieter Lang, Aug 08 2012

nonn

This is the fifth column (m=5) of triangle A214306.
Each 5 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[5]], with p[1] >= p[2] >= .. >=  p[5] >= 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,5)= A008284(n,5) partitions of n with 5 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
It appears that this sequence is divisible by 12, producing  1, 75, 2100, 36890, 487200, 5368356, 51847950, 455513850, ...
Compare this with A056345 where only 5 colors are used for all n >= 5.

			a(6) = A213941(6,10) = 900 from the bracelet with color signature [2,1,1,1,1] and color repertoire [c[j], j=1, 2, ..., 6]. There are A213939(6,10) = 30 bracelets with representative color multinomials c[1]^2 c[2] c[3] c[4] c[5]. If the colors c[j] are taken as j, e.g., 112345, 112354, 112435, 112453, 112534, 112543, 113245, 113254, 113425, (113452 is equivalent to 112543 by turning over), 113524, (113542 ==112453), 114235, ..., 121345, ... (all taken cyclically). Each of these 30 bracelets represents a class of A035206(6,10) = 30 bracelets when all six colors are used. Thus a(6) = 30*30 = 900 = 12*75.

Andrew Howroyd, Table of n, a(n) for n = 5..100

Cf. A213941, A214306, A214311 (m=5, representative bracelets), A214312 (m=4).

a(n) = A214306(n,5), n >= 5.
a(n) = sum(A213941(n,k),k = A214314(n,5) .. (A214314(n,5) - 1 + A008284(n,5))), n >= 5.
a(n) = binomial(n,5) * A056345(n). - Andrew Howroyd, Mar 25 2017

A292223 a(n) is the number of representative six-color bracelets (necklaces with turning over allowed; D_6 symmetry) with n beads, for n >= 6.

Original entry on oeis.org

60, 180, 1050, 5040, 29244, 161340, 1046250, 4825800, 27790266, 145126548, 843333015, 4466836920, 26967624184, 137243187108, 789854179074, 4306147750200, 24711052977222, 134216193832908, 797987818325009, 4240082199867228

Offset: 6

Wolfdieter Lang, Sep 30 2017

nonn

This is the sixth column (m = 6) of triangle A213940.
The relevant p(n,6)= A008284(n, 6) representative color multinomials have exponents (signatures) from the six-part partitions of n, written with nonincreasing parts. E.g., n = 8: [3,1,1,1,1,1] and [2,2,1,1,1,1] (p(8,6)=2). The corresponding representative bracelets have the six-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5]*c[6] and c[1]^2*c[2]^2*c[3]*c[4]*c[5]*c[6].
See A056361 for the numbers if also color permutations for D_6 inequivalent bracelets are allowed. (Andrew Howroyd induced me to look at these bracelets.)

			a(6) = A213940(6,6) = A213939(6, 11) = 60 from the representative bracelets (with colors j for c(j), j=1..6) permutations of (1, 2, 3, 4, 5, 6) modulo D_6 (dihedral group) symmetry, i.e., modulo cyclic or anti-cyclic operations. E.g., (1, 2, 3, 4, 6, 5) == (2, 3, 4, 6, 5, 1) == (6, 4, 3, 2, 1, 5) == ..., but (1, 2, 3, 4, 6, 5) is not equivalent to (1, 2, 3, 4, 5, 6). If color permutation is also allowed, then there is only one possibility (see A056361(6) = 1).

Cf. A008284, A036036, A056361, A213940, A214311, A214314.

a(n) = A213940(n, 6), n >= 6.

a(n) = Sum_{k=b(n, 6)..b(n, 7)-1} A213939(n, k), for n >= 7, 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), and a(6) = A213939(6, b(6,6)) = A213939(6, 11) = 60.

cp's OEIS Frontend

A213940 Triangle with entry a(n,m) giving the number of bracelets of n beads (dihedral D_n symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.

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A214309 a(n) is the number of representative four-color bracelets (necklaces with turning over allowed) with n beads, for n >= 4.

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A056360 Number of bracelet structures using exactly five different colored beads.

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A214313 a(n) is the number of all five-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 5.

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A292223 a(n) is the number of representative six-color bracelets (necklaces with turning over allowed; D_6 symmetry) with n beads, for n >= 6.

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