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A213936
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Number triangle with entry a(n,k), n>=1, m=1, 2, ..., n, giving the number of representative necklaces with n beads (C_n symmetry) corresponding to the color multinomial c[1]^k*c[2]*...*c[n+1-k].
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6
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1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 24, 12, 4, 1, 1, 120, 60, 20, 5, 1, 1, 720, 360, 120, 30, 6, 1, 1, 5040, 2520, 840, 210, 42, 7, 1, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1
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OFFSET
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1,4
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COMMENTS
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This table coincides with A173333 but has an extra main diagonal with entries 1.
a(n,k) is the number of necklaces of n beads (C_N symmetry), with colors from the repertoire {c[1],c[2],...,c[n]}, corresponding to the representative color multinomials obtained from the partition [k,1^(n-k)] of n with m=n-k+1 parts by 'exponentiation' (taking the parts in the given order as exponents of the colors), hence only m from the available n colors are present. As representative necklaces one takes the ones where the color c[1] appears k times. In particular, for k=1 the partition is [1^n] and all n colors are used, and there are (n-1)! necklaces from permuting the n colors.
a(n,k) appears in the representative necklace partition array A212359 in row n at the position l(n,n+1-k,1), with l(n,m,1) the position of the first partition with m parts in the list of partitions of n in A-St order. E.g., n=5, k=4: l(5,5-3,1) =2 with the partition [4,1] (used in reverse order compared to A-St).
See the comments on A212359 for the Abramowitz-Stegun (A-St) reference, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions.
The row sums of this triangle are given by A213937.
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LINKS
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FORMULA
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a(n,n)=1, a(n,k) = (n-1)!/k! if 1 <= k < n, else 0.
See also A212359 with a link for the formula for general partitions.
a(n,k) = A173333(n-1,k), 1 <= k < n.
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EXAMPLE
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n\k 1 2 3 4 5 6 7 8 9 10 ...
1 1
2 1 1
3 2 1 1
4 6 3 1 1
5 24 12 4 1 1
6 120 60 20 5 1 1
7 720 360 120 30 6 1 1
8 5040 2520 840 210 42 7 1 1
9 40320 20160 6720 1680 336 56 8 1 1
10 362880 181440 60480 15120 3024 504 72 9 1 1 ...
a(4,3) = 1 because the partition is [3,1], the color signature (exponentiation) c[.]^3 c[.]^1, and the one representative necklace (we use j for color c[j] here) is: cyclic(1112).
a(4,2) = 3 because the partition is [2,1^2], the color signature c[.]^2 c[.] c[.], and the three representative necklaces are: cyclic(1123), cyclic(1132) and cyclic(1213).
a(5,3) = 4 because the color signature is c[.]^3 c[.] c[.] (from the partition [3,1^2]). and the four representative necklaces are 11123, 11132, 11213 and 11312, all taken cyclically.
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CROSSREFS
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Cf. A212359, A213937 (row sums). For columns and diagonals see the links under A173333 (after an additional 1 has been supplied for each columns).
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KEYWORD
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AUTHOR
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STATUS
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approved
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