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A228094 Triangle starting at row 3 read by rows of the number of permutations in the n-th Dihedral group which are the product of k disjoint cycles, d(n,k), n >= 3, 1 <= k <= n. 0
2, 3, 1, 2, 3, 2, 1, 4, 0, 5, 0, 1, 2, 2, 4, 3, 0, 1, 6, 0, 0, 7, 0, 0, 1, 4, 2, 0, 5, 4, 0, 0, 1, 4, 2, 0, 5, 4, 0, 0, 1, 6, 0, 2, 0, 9, 0, 0, 0, 1, 6, 0, 2, 0, 9, 0, 0, 0, 1, 4, 4, 0, 0, 6, 5, 0, 0, 0, 1, 10, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 7, 6, 0, 0, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

The multivariable row polynomials give n times the cycle index for the Dihedral group D_n, called Z(D_n) (see the MathWorld link with the Harary reference). For example, 12*Z(D_6) = 2*(y_6)^1 + 2*(y_3)^2 + 4*(y_2)^3+3*(y_1)^2*(y_2)^2 + 1*(y_1)^6.

REFERENCES

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, 1973, p. 37.

LINKS

Table of n, a(n) for n=3..94.

Eric W. Weisstein, MathWorld: Cycle Index

FORMULA

d(n,k) = phi(n/k) + d'(n,k), where: If n is odd, then d'(n,k)= n when k=(n+1)/2 and d'(n,k)=0 otherwise. If n is even, then d'(n,k)=n/2 when k=n/2, (n+2)/2 and d'(n,k)=0 otherwise.

EXAMPLE

Triangle begins

2,  3, 1;

2,  3, 2, 1;

4,  0, 5, 0, 1;

2,  2, 4, 3, 0, 1;

6,  0, 0, 7, 0, 0,  1;

4,  2, 0, 5, 4, 0,  0, 1;

6,  0, 2, 0, 9, 0,  0, 0, 1;

4,  4, 0, 0, 6, 5,  0, 0, 0, 1;

10, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1;

4,  2, 2, 2, 0, 7,  6, 0, 0, 0, 0, 1;

CROSSREFS

Cf. A054523, A130534, A000010.

Sequence in context: A054073 A194871 A194899 * A059832 A105316 A105933

Adjacent sequences:  A228091 A228092 A228093 * A228095 A228096 A228097

KEYWORD

nonn,tabf

AUTHOR

Robert A. Beeler, Aug 09 2013

STATUS

approved

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Last modified January 17 20:06 EST 2022. Contains 350410 sequences. (Running on oeis4.)