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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.

This array is obtained by multiplying the entry of the array A213939(n,k) (number of bracelets (dihedral D_n symmetry) with n beads, each available in n colors, with color representative given by the n-multiset representative obtained from the k-th partition of n in A-St order after 'exponentiation') with the entry of the array A035206(n,k) (number of members in the equivalence class represented by the color multiset considered for A213939(n,k)): a(n,k)=A213939(n,k)*A035206(n,k), k=1..p(n)=A000041(n), n>=1. The row sums then give the total number of bracelets with n beads from n colors, given by A081721(n).

See A212359 for references, the 'exponentiation', and a link. For multiset signatures and representative multisets defining color multinomials see also a link in A213938.

The corresponding triangle with the summed row entries related to partitions of n with fixed number of parts is A214306.

This triangle is obtained from the array A213941 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 total number of necklaces of n beads (dihedral D_n symmetry) corresponding to all the color multinomials obtained from all p(n,m) = A008284(n,m) partitions of n with m parts, written in nonincreasing form, by 'exponentiation'. Therefore only m from the available n colors are present, and a(n,m) gives the number of bracelets with n beads with only m of the n available colors present, for m from 1,2,...,n, and n >= 1. All of the possible color assignments are counted.

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. See a link in A213938 for representative multisets for given signature used to define color multinomials.

The row sums of this triangle coincide with the ones of array A213941, and they are given by A081721.