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A212360
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Partition array a(n,k) with the total number of necklaces (C_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.
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4
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1, 2, 1, 3, 6, 2, 4, 12, 12, 36, 6, 5, 20, 40, 120, 180, 240, 24, 6, 30, 90, 60, 300, 1200, 320, 1200, 2700, 1800, 120, 7, 42, 126, 210, 630, 3150, 2100, 3150, 4200, 25200, 12600, 12600, 37800, 15120, 720, 8, 56, 224, 392, 280, 1176, 7056, 11760, 9072, 11760, 11760, 88200, 58800, 176400, 22260, 58800, 470400, 352800, 141120, 529200, 141120, 5040
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OFFSET
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1,2
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COMMENTS
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This array is obtained by multiplying the entry of the array A212359(n,k) (number of necklaces (C_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 A212359(n,k)): a(n,k)=A212359(n,k)* A035206(n,k), k=1..p(n)= A000041(n), n>=1. The row sums then give the total number of necklaces with beads from n colors, given by A056665(n).
See A212359 for references, the 'exponentiation', and a link.
The corresponding triangle with the summed row entries which belong to partitions of n with fixed number of parts is A213935. [From Wolfdieter Lang, Jul 12 2012]
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LINKS
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FORMULA
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EXAMPLE
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n\k 1 2 3 4 5 6 7 8 9 10 11
1 1
2 2 1
3 3 6 2
4 4 12 12 36 6
5 5 20 40 120 180 240 24
6 6 30 90 60 300 1200 320 1200 2700 1800 120
...
See the link for the rows n=1..15.
a(3,1)=3 because the 3 necklaces with 3 beads coming in 3 colors have the color multinomials (here monomials) c[1]^3=c[1]*c[1]*c[1], c[2]^3 and c[3]^3. The partition of 3 is 3, the color representative is c[1]^3, and the equivalence class with color signature from the partition 3 has the three given members.
a(3,2)=6 from the color signature 2,1 with the representative multinomial c[1]^2 c[2] with coefficient A212359(3,2)=1, the only 3-necklace cyclic(112) (taking j for the color c[j]), and A035206(3,2)=6 members of the whole color equivalence class: cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332).
a(3,3)=2, color signature 1^3=1,1,1 with representative multinomial c[1]*c[2]*c[3] with coefficient A212359(3,3)=2 from the two necklaces cyclic(1,2,3) and cyclic (1,3,2). There are no other members in this class (A035206(3,3)=1).
The sum of row nr. 3 is 11=A056665(3). See the example given there with c[1]=R, c[2]=G and c[3]=B.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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