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%I #31 Jul 06 2023 21:09:16
%S 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,
%T 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,
%U 0,7,6,0,0,0,0,1,12,0,0,0,0,0,13,0,0,0,0,0,1
%N 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.
%C 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.
%D Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015. See Theorem 8.4.12 at pp. 246-247.
%D Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, 1973, p. 37.
%H Stefano Spezia, <a href="/A228094/b228094.txt">First 150 rows of the triangle, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleIndex.html">Cycle Index</a>.
%F d(n,k) = A054523(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.
%e Triangle begins
%e 2, 3, 1;
%e 2, 3, 2, 1;
%e 4, 0, 5, 0, 1;
%e 2, 2, 4, 3, 0, 1;
%e 6, 0, 0, 7, 0, 0, 1;
%e 4, 2, 0, 5, 4, 0, 0, 1;
%e 6, 0, 2, 0, 9, 0, 0, 0, 1;
%e 4, 4, 0, 0, 6, 5, 0, 0, 0, 1;
%e 10, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1;
%e 4, 2, 2, 2, 0, 7, 6, 0, 0, 0, 0, 1;
%e ...
%t d[n_,k_]:=If[Divisible[n,k],EulerPhi[n/k],0]+If[OddQ[n]&&k==(n+1)/2,n,If[EvenQ[n]&&(k==n/2||k==(n+2)/2),n/2,0]]; Table[d[n,k],{n,3,12},{k,n}]//Flatten (* _Stefano Spezia_, Jun 26 2023 *)
%Y Cf. A000010, A054523, A130534.
%K nonn,tabf
%O 3,1
%A _Robert A. Beeler_, Aug 09 2013
%E Terms corrected by _Stefano Spezia_, Jun 30 2023
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