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%I #21 Feb 02 2022 20:36:52
%S 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,
%T 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,
%U 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
%N Coefficients for the cycle index polynomial for the dihedral group D_n multiplied by 2n, n>=1, read as partition polynomial.
%C 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).
%C 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.
%C 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.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 37, (2.2.11).
%H Andrew Howroyd, <a href="/A212355/b212355.txt">Table of n, a(n) for n = 1..2713</a> (rows 1..20)
%H Wolfdieter Lang, <a href="/A212355/a212355.pdf">Cycle index polynomials Z(D_n), n=1..15</a>
%F 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.
%F 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).
%e n\k 1 2 3 4 5 6 7 8 9 10 11 ...
%e 1: 2
%e 2: 2 2
%e 3: 2 3 1
%e 4: 2 0 3 2 1
%e 5: 4 0 0 0 5 0 1
%e 6: 2 0 0 2 0 0 4 0 3 0 1
%e ...
%e See the link for rows n=1..8 and the corresponding Z(D_n) polynomials for n=1..15.
%e 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
%o (PARI)
%o 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)) }
%o row(n)=[C(Vec(p)) | p<-Vec(partitions(n))]
%o { for(n=1, 7, print(row(n))) } \\ _Andrew Howroyd_, Feb 02 2022
%Y Cf. A054523, A102190, A212356.
%K nonn,tabf
%O 1,1
%A _Wolfdieter Lang_, Jun 02 2012
%E Terms a(67) and beyond from _Andrew Howroyd_, Feb 02 2022
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