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The row lengths sequence is A000041(n), n >= 1.

The partitions are ordered like in Abramowitz-Stegun (A-St order). For the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used.

A bracelet with n beads (n-bracelet) has the dihedral D_n symmetry group of degree n (order 2n). In addition to cyclic C_n operations, also a turnover (in 3-space) or a reflection (in 2-space) is allowed. In the Harary-Palmer reference, p. 44, the term necklace is used instead of bracelet.

a(n,k) gives the number of representative n-bracelets, with up to n colors for each bead, belonging to the k-th partition of n in A-St order in the following way. Write this partition with nonincreasing parts (this is the reverse of the partition as given by A-St), e.g., [3,1^2], not [1^2,3], which is written as [3,1,1], a partition of n=5. In general a (reversed) partition of n is written as [p[1],p[2],...,p[m]], with p[1] >= p[2] >= ... >= p[m] >= 1, with m the number of parts. To each such partition of n corresponds an n-multiset obtained by 'exponentiation'. For more details see the W. Lang link in A213938 with more details as well as a list of multiset signatures and corresponding multiset representatives. For the given example the 5-multiset is {1^3,2^1,3^1}={1,1,1,2,3}. In general, {1^p[1],2^p[2],...,m^p[m]}. We will also use a list notation with square brackets for these multisets. Such an n-multiset representative (of a repetition class defined by the exponents, also called signature) encodes the representative n-bracelet color monomial by c[1]^p[1]*c[2]^p[2]*...*c[m]^p[m]. For the example one has c[1]^3*c[2]*c[3]. The number of 5-bracelets with this color assignment is a(5,4) because [3,1,1] is the 4th partition of 5 in A-St order. The a(5,4)=2 non-equivalent 5-bracelets with this color assignment are cyclic(c[1]c[1]c[1]c[2]c[3]) and cyclic(c[1]c[1]c[2]c[1]c[3]). For the necklace case c[1]c[1]c[1]c[3]c[2] and c[1]c[1]c[3]c[1]c[2] (both taken cyclically) also have to be counted, but due to a turn over (or a reflection) they become equivalent to the two given bracelets, respectively.

Such a set of a(n,k) n-bracelets for the given color signature stands for other sets of the same order when different colors from the repertoire {c[1],...,c[n]} are chosen. In the example, the partition [3,1,1] with the representative multiset [1^3,2,3] stands for all-together 5*binomial(4,2) = 30 such sets, each leading to 2 possible non-equivalent 5-bracelet arrangements. Thus one has all-together 30*2=60 5-bracelets with color signature determined from the partition [3,1,1]. See the partition array A213941 for these total bracelet numbers.

a(n,k) is computed from the cycle index Z(D_n) for the dihedral group (see A212355 and the link given there) after the variables x_j have been replaced by the j-th power sum sum(c[i]^j,i=1..n), abbreviated as Z(D_n,c_n) with c_n:=sum(c[i],i=1..n), n >= 1. The coefficient of the representative color multinomial determined by the k-th partition of n in A-St order, as explained above, is a(n,k). See the Harary-Palmer reference, p. 36, Theorem (PET) with A = D_n and p. 37 eq. (2.2.11) for the cycle index polynomial Z(D_n). See the W. Lang link for more details.

The row sums are given by A213943.

a(n) is the coefficient of c_1^n*c_2^n in the cycle index polynomial for the dihedral group D_{2*n} evaluated with the figure counting polynomial c = c_1 + c_2, n>=1, abbreviated as Z(D_{2*n},c). See, e.g., the Harary-Palmer reference (given under A212355), p. 42, Theorem (PET), and the example for all 6 two-colored 4-bracelets (called there necklaces) on p. 44, Figure 2.4.2. - Wolfdieter Lang, Jun 05 2012

See A212355 for the formula for the cycle index Z(D_n) of the dihedral group, the Harary and Palmer reference, and a link for these polynomials for n=1..15.

It seems that this is also the number of different sets of distances of n points placed on 2n equidistant points on a circle. - M. F. Hasler, Jan 28 2013

The row lengths sequence is A000041.

The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used).

The row sums are A001710(n-1), n>=1.

The cycle index (multivariate polynomial) for the alternating group A_n, called Z(A_n), is

Z(S_n) + Z(S_n;x[1],-x[2],x[3],-x[4],... ), n>=1,

with the cycle index Z(S_n) for the symmetric group S_n, in the variables x[1],...,x[n]. See the Harary and Palmer reference. The coefficients of n!*Z(S_n) are the M_2 numbers of Abramowitz-Stegun, pp. 831-2. See A036039 and A102189, also for the Abramowitz-Stegun reference.