%I
%S 1,2,1,3,6,2,4,12,12,36,6,5,20,40,120,180,240,24,6,30,90,60,300,1200,
%T 320,1200,2700,1800,120,7,42,126,210,630,3150,2100,3150,4200,25200,
%U 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
%N 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 kth partition of n in AbramowitzStegun(ASt) order.
%C 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 nmultiset representative obtained from the kth partition of n in ASt 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).
%C See A212359 for references, the 'exponentiation', and a link.
%C 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]
%H Wolfdieter Lang, <a href="/A212360/a212360.pdf">Rows n=1..15.</a>
%F a(n,k) = A212359(n,k)*A035206(n,k), k=1,2,...,p(n)= A000041(n), n>=1.
%e n\k 1 2 3 4 5 6 7 8 9 10 11
%e 1 1
%e 2 2 1
%e 3 3 6 2
%e 4 4 12 12 36 6
%e 5 5 20 40 120 180 240 24
%e 6 6 30 90 60 300 1200 320 1200 2700 1800 120
%e ...
%e See the link for the rows n=1..15.
%e 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.
%e 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 3necklace 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).
%e 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).
%e 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.
%Y Cf. A212359, A035206, A056665, A213935.
%K nonn,tabf
%O 1,2
%A _Wolfdieter Lang_, Jun 25 2012
