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A212357 Coefficients for the cycle index polynomial for the cyclic group C_n multiplied by n, n>=1, read as partition polynomial.

%I A212357 #9 Jun 12 2012 17:31:57
%S A212357 1,1,1,2,0,1,2,0,1,0,1,4,0,0,0,0,0,1,2,0,0,2,0,0,1,0,0,0,1,6,0,0,0,0,
%T A212357 0,0,0,0,0,0,0,0,0,1,4,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,
%U A212357 0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0
%N A212357 Coefficients for the cycle index polynomial for the cyclic group C_n multiplied by n, n>=1, read as partition polynomial.
%C A212357 The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used).
%C A212357 The row lengths sequence is A000041. The number of nonzero entries in row nr. n is  A000005(n).
%C A212357 The cycle index (multivariate polynomial) for the cyclic group C_n, called Z(C_n), is (sum(phi(k)*x_k^{n/k} ,k divides n))/n, n>=1, with Euler's totient function phi(n)= A000010(n). See the Harary and Palmer reference. For the coefficients of Z(C_n) in different tabulations see also A054523 and A102190.
%D A212357 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).
%H A212357 Wolfdieter Lang, <a href="/A212357/a212357.pdf">Cycle index Z(C_n), n=1..15.</a>
%F A212357 The cycle index polynomial for the cyclic group C_n is Z(C_n) = (a(n,k)*x[1]^(e[k,1])*x[2]^(e[k,2])*...*x[n]^(e[k,n]))/n, n>=1, if the k-th partition of n in Abramowitz-Stegun order is 1^(e[k,1]) 2^(e[k,2]) ... n^(e[k,n]), where a part j with vanishing exponent e[k,j] has to be omitted. The n dependence of the exponents has been suppressed. See the comment above for the Z(C_n) formula and the link for these polynomials for n=1..15.
%F A212357 a(n,k) is the coefficient the term of n*Z(C_n) corresponding to the k-th partition of n in Abramowitz-Stegun order. a(n,k) = 0 if there is no such term in Z(C_n).
%e A212357 n\k  1 2 3 4 5 6 7 8 9 10 11 ...
%e A212357 1:   1
%e A212357 2:   1 1
%e A212357 3:   2 0 1
%e A212357 4:   2 0 1 0 1
%e A212357 5:   4 0 0 0 0 0 1
%e A212357 6:   2 0 0 2 0 0 1 0 0  0  1
%e A212357 ...
%e A212357 See the link for rows n=1..8 and the Z(C_n) polynomials for n=1..15.
%e A212357 n=6: Z(C_6) = (2*x[6] + 2*x[3]^2 + 1*x[2]^3 + x[1]^6)/6, because the relevant partitions of 6 appear for k=1: 6, k=4: 3^2, k=7: 2^3 and k=11: 1^6
%Y A212357 Cf. A000005, A000010, A054523, A102190.
%K A212357 nonn,tabf
%O A212357 1,4
%A A212357 _Wolfdieter Lang_, Jun 04 2012