By George Pólya; Ronald C Read
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Extra info for Combinatorial enumeration of groups, graphs, and chemical compounds
Now we turn to arbitrary planted trees with a total_of n nodes: Tn denotes the number of topologicals different trees, P n the number of two�dimensionally different trees. In addition to the generating function (2), we consider ~p~(x) » Pxx + P2x2 + P s x s + • • • . 25) TX = P X = 1. For n > 2 the planted tree has principal branches; let к � 1 be their number, as in Sec. 38. These к � 1 principal branches contain a total of n � I nodes, the subgroup associated with their configuration is $ k _!
In addition to the generating function (2), we consider ~p~(x) » Pxx + P2x2 + P s x s + • • • . 25) TX = P X = 1. For n > 2 the planted tree has principal branches; let к � 1 be their number, as in Sec. 38. These к � 1 principal branches contain a total of n � I nodes, the subgroup associated with their configuration is $ k _! , depending on whether 7"n or P n is involved. The number" of configurations of principal branches which are non� equivalent with respect to $k_x is, according to Sec. 16, the n_1 in the series which obtains upon substitution of coefficient of x t(x) in the cycle index of $k_x.
1 Denoting by Aj, A2 AT the orders of the groups of automorphisms which correspond to the т different trees, we have L L L. 40). 2 (b) Jordan indicated a method to determine the order of the group of automorphisms of an arbitrary graph. , $m by repeated application of the two in Sec. 27 discussed operations: the direct product G * Я and the corona construction of С with respect to H , 6[ H]. In particular, the order of the group of automorphisms has to assume the form a. a. a mx\ m2\ ... w_! , a r are some natural numbers.