

A212357


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


3



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

1,4


COMMENTS

The partitions are ordered like in AbramowitzStegun (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.


REFERENCES

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


LINKS

Table of n, a(n) for n=1..87.
Wolfdieter Lang, Cycle index Z(C_n), n=1..15.


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 kth partition of n in AbramowitzStegun 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 kth partition of n in AbramowitzStegun order. a(n,k) = 0 if there is no such term in Z(C_n).


EXAMPLE

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


CROSSREFS

Cf. A000005, A000010, A054523, A102190.
Sequence in context: A115723 A238160 A178524 * A114525 A127672 A294168
Adjacent sequences: A212354 A212355 A212356 * A212358 A212359 A212360


KEYWORD

nonn,tabf


AUTHOR

Wolfdieter Lang, Jun 04 2012


STATUS

approved



