A212355 Coefficients for the cycle index polynomial for the dihedral group D_n multiplied by 2n, n>=1, read as partition polynomial.

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, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Wolfdieter Lang, Jun 02 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 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.

Examples

			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
		

References

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..2713 (rows 1..20)
• Wolfdieter Lang, Cycle index polynomials Z(D_n), n=1..15

Crossrefs

Cf. A054523, A102190, A212356.

Programs

PARI

C(v)={my(n=vecsum(v), r=#v); if(v[1]==v[r], eulerphi(v[1])) + if(v[r]<=2 && 2*r <= n+2, if(n%2, n, n/2)) }
row(n)=[C(Vec(p)) | p<-Vec(partitions(n))]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022

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).

Extensions

Terms a(67) and beyond from Andrew Howroyd, Feb 02 2022