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A212357
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Coefficients for the cycle index polynomial for the cyclic group C_n multiplied by n, n>=1, read as partition polynomial.
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3
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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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,4
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COMMENTS
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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).
The row lengths sequence is A000041. The number of nonzero entries in row nr. n is A000005(n).
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.
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REFERENCES
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F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).
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LINKS
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FORMULA
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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.
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).
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EXAMPLE
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n\k 1 2 3 4 5 6 7 8 9 10 11 ...
1: 1
2: 1 1
3: 2 0 1
4: 2 0 1 0 1
5: 4 0 0 0 0 0 1
6: 2 0 0 2 0 0 1 0 0 0 1
...
See the link for rows n=1..8 and the Z(C_n) polynomials for n=1..15.
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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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