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A212355 Coefficients for the cycle index polynomial for the dihedral group D_n multiplied by 2n, n>=1, read as partition polynomial. 6
2, 2, 2, 2, 3, 1, 2, 0, 3, 2, 1, 4, 0, 0, 0, 5, 0, 1, 2, 0, 0, 2, 0, 0, 4, 0, 3, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 1, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 4, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 1 for n=1, 2 for n=2 and A000005(n)+1 otherwise. This is the sequence A212356.

The cycle index (multivariate polynomial) for the dihedral group D_n (of order 2*n), called Z(D_n), is for odd n given by (Z(C_n) + x[1]*x[2]^((n-1)/2))/2 and for even n by (2*Z(C_n) + x[2] ^(n/2) + x[1]^2*x[2]^((n-2)/2))/4, where Z(C_n) is the cycle index for the cyclic group C_n. For the coefficients of Z(C_n) see A054523 or A102190. See the Harary and Palmer reference.

REFERENCES

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 37, (2.2.11).

LINKS

Table of n, a(n) for n=1..66.

Wolfdieter Lang, Cycle index polynomials Z(D_n), n=1..15

FORMULA

The cycle index polynomial for the dihedral group D_n is Z(D_n) = (a(n,k)*x[1]^(e[k,1])*x[2]^(e[k,2])*...*x[n]^(e[k,n]))/(2*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(D_n) formula and the link for these polynomials for n=1..15.

a(n,k) is the coefficient the term of 2*n*Z(D_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(D_n).

EXAMPLE

n\k   1  2  3  4  5  6  7  8  9 10 11 ...

1:    2

2:    2  2

3:    2  3  1

4:    2  0  3  2  1

5:    4  0  0  0  5  0  1

6:    2  0  0  2  0  0  4  0  3  0  1

...

See the link for rows n=1..8 and the corresponding Z(D_n) polynomials for n=1..15.

n=6: Z(D_6) = (2*x[6] + 2*x[3]^2 +  4*x[2]^3 + 3*x[1]^2*x[2]^2 + x[1]^6)/12, 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. A054523, A102190, A212356.

Sequence in context: A090872 A283472 A225538 * A238646 A194330 A280667

Adjacent sequences:  A212352 A212353 A212354 * A212356 A212357 A212358

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Jun 02 2012

STATUS

approved

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Last modified February 16 20:45 EST 2019. Contains 320189 sequences. (Running on oeis4.)