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 A213941 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. 9
 1, 2, 1, 3, 6, 1, 4, 12, 12, 24, 3, 5, 20, 40, 60, 120, 120, 12, 6, 30, 90, 45, 180, 720, 220, 600, 1440, 900, 60, 7, 42, 126, 168, 315, 1890, 1050, 1890, 2100, 12600, 6720, 6300, 18900, 7560, 360, 8, 56, 224, 280, 224, 672, 4032, 6384, 5544, 6384, 5880, 45360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Wolfdieter Lang, Rows n=1 to n=15. FORMULA a(n,k) = A213939(n,k)*A035206(n,k), k=1, 2, ..., p(n) = A000041(n), n >= 1. EXAMPLE n\k 1   2    3    4    5     6     7     8     9    10   11 1   1 2   2   1 3   3   6    1 4   4  12   12   24    3 5   5  20   40   60  120   120    12 6   6  30   90   45  180   720   220   600  1440   900   60 ... Row m=7 is: 7 42 126 168 315 1890 1050 1890 2100 12600 6720 6300 18900 7560 360. For the rows n=1 to n=15 see the link. a(3,1) = 3 because the 3 bracelets with 3 beads coming in 3 colors have the color multinomials (here monomials) c^3=c*c*c, c^3 and c^3. The partition of 3 is , the color representative is c^3, and the equivalence class with color signature from the partition  has the three given members. There is no difference between necklace and bracelet numbers in this case. a(3,2) = 6 from the color signature 2,1 with the representative multinomial c^2 c 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. a(3,3) = 1, color signature 1^3 = 1,1,1 with representative multinomial c*c*c 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). 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. CROSSREFS Cf. A213939, A035206, A081721, A214306. Sequence in context: A035206 A210238 A209936 * A181511 A115196 A093346 Adjacent sequences:  A213938 A213939 A213940 * A213942 A213943 A213944 KEYWORD nonn,tabf AUTHOR Wolfdieter Lang, Jul 20 2012 STATUS approved

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Last modified August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)