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A212355
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Coefficients for the cycle index polynomial for the dihedral group D_n multiplied by 2n, n>=1, read as partition polynomial.
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6
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 37, (2.2.11).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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
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PROG
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(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))]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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