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%I #10 Aug 01 2012 05:07:50
%S 1,2,1,3,6,1,4,12,12,24,3,5,20,40,60,120,120,12,6,30,90,45,180,720,
%T 220,600,1440,900,60,7,42,126,168,315,1890,1050,1890,2100,12600,6720,
%U 6300,18900,7560,360,8,56,224,280,224,672,4032,6384,5544,6384,5880,45360
%N Partition array a(n,k) with the total number of bracelets (D_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.
%C 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).
%C See A212359 for references, the 'exponentiation', and a link. For multiset signatures and representative multisets defining color multinomials see also a link in A213938.
%C The corresponding triangle with the summed row entries related to partitions of n with fixed number of parts is A214306.
%H Wolfdieter Lang, <a href="/A213941/a213941.pdf">Rows n=1 to n=15.</a>
%F a(n,k) = A213939(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 1
%e 4 4 12 12 24 3
%e 5 5 20 40 60 120 120 12
%e 6 6 30 90 45 180 720 220 600 1440 900 60
%e ...
%e Row m=7 is: 7 42 126 168 315 1890 1050 1890 2100 12600 6720 6300 18900 7560 360.
%e For the rows n=1 to n=15 see the link.
%e a(3,1) = 3 because the 3 bracelets 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. There is no difference between necklace and bracelet numbers in this case.
%e a(3,2) = 6 from the color signature 2,1 with the representative multinomial c[1]^2 c[2] with coefficient A213939(3,2) = 1, the only 3-bracelet 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). There is no difference between necklaces and bracelets numbers in this case.
%e a(3,3) = 1, color signature 1^3 = 1,1,1 with representative multinomial c[1]*c[2]*c[3] with coefficient A213939(3,3)=1 from the bracelet cyclic(1,2,3). The necklace (1,3,2) becomes equivalent to this one under D_3 operation. There are no other members in this class (A035206(3,3)=1).
%e The sum of row No. 3 is 10 = A081721(3). The bracelets are 111, 222, 333, 112, 113, 221, 223, 331, 332 and 123, all taken cyclically.
%Y Cf. A213939, A035206, A081721, A214306.
%K nonn,tabf
%O 1,2
%A _Wolfdieter Lang_, Jul 20 2012
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