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

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

Offset: 1

Author

Wolfdieter Lang, Jun 04 2012

Keywords

• nonn
• tabf

Comments

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.

Examples

			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
		

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).

Links

• Wolfdieter Lang, Cycle index Z(C_n), n=1..15.

Crossrefs

Cf. A000005, A000010, A054523, A102190.

Formula

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).