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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].

%I A213936 #10 Jan 28 2013 04:25:36
%S A213936 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,
%T A213936 5040,2520,840,210,42,7,1,1,40320,20160,6720,1680,336,56,8,1,1,362880,
%U A213936 181440,60480,15120,3024,504,72,9,1,1
%N 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].
%C A213936 This table coincides with A173333 but has an extra main diagonal with entries 1.
%C A213936 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.
%C A213936 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).
%C A213936 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.
%C A213936 The row sums of this triangle are given by A213937.
%F A213936 a(n,n)=1, a(n,k) = (n-1)!/k! if 1 <= k < n, else 0.
%F A213936 See also A212359 with a link for the formula for general partitions.
%F A213936 a(n,k) = A173333(n-1,k), 1 <= k < n.
%e A213936 n\k      1       2      3      4     5    6   7  8  9 10 ...
%e A213936 1        1
%e A213936 2        1       1
%e A213936 3        2       1      1
%e A213936 4        6       3      1      1
%e A213936 5       24      12      4      1     1
%e A213936 6      120      60     20      5     1    1
%e A213936 7      720     360    120     30     6    1   1
%e A213936 8     5040    2520    840    210    42    7   1  1
%e A213936 9    40320   20160   6720   1680   336   56   8  1  1
%e A213936 10  362880  181440  60480  15120  3024  504  72  9  1  1 ...
%e A213936 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).
%e A213936 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).
%e A213936 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.
%Y A213936 Cf. A212359, A213937 (row sums). For columns and diagonals see the links under A173333 (after an additional 1 has been supplied for each columns).
%K A213936 nonn,easy,tabl
%O A213936 1,4
%A A213936 _Wolfdieter Lang_, Jul 10 2012