A212360 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.

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

Offset: 1

Author

Wolfdieter Lang, Jun 25 2012

Keywords

• nonn
• tabf

Comments

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]

Examples

			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.
		

Links

• Wolfdieter Lang, Rows n=1..15.

Crossrefs

Cf. A212359, A035206, A056665, A213935.

Formula

a(n,k) = A212359(n,k)*A035206(n,k), k=1,2,...,p(n)= A000041(n), n>=1.