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A213936 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]. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
FORMULA
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.
EXAMPLE
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.
CROSSREFS
Cf. A212359, A213937 (row sums). For columns and diagonals see the links under A173333 (after an additional 1 has been supplied for each columns).
Sequence in context: A201922 A181644 A144351 * A142589 A284308 A369435
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Jul 10 2012
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)