Combinatorial enumeration of groups, graphs, and chemical by George Pólya; Ronald C Read

By George Pólya; Ronald C Read

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The tetrahedron vertices are numbered so that the tetrahedron carries the positive orientation. This means that a person, with the head in vertex 1 and the feet in vertex 2, facing edge 3 � 4 has 3 on the left and 4 on the right�hand side. In this way we can order the edges of the corona in 12 ways. It is easy to see that an even permutation of the vertices leaves the numbering "right�handed", an odd one produces a "left�handed" 1 numbering. Thus the 12 different numbering schemes are mapped into each other by the 12 permutations of the alternating group of degree 4, A4.

Groups 27. We discuss some cases in which the cycle index of a group composed of several groups can be constructed from the cycle indices of the given groups in a transparent way. Let G and H be two permutation groups with, respectively, orders g and h, degrees г and s, and cycle indices

We use С a n d H to construct two new permutation groups. T h e first is very simple a n d well�known, the second is more interesting.

For n > 2 the planted C-tree has 1, 2, or 3 principal branches. 26) leads for the series Rxx + R2x2 + Rzxz + ... 30) g(x) m x + xg(x) + x(g(x)2 + g(x2))/2 + x(g(x)z + 3g(x)g(x2) + 2¿Kxs))/6. 30) contains only four terms; they correspond to the four possible cases of a rooted C-tree: there are 0, 1, 2, or 3 principal branches. 30), we obtain equation (4) as expected. 1 45. Now we examine the structure of ( l ' ) and ( I м ) of equation (1) from a different angle. We consider free trees with n mutually different vertices; a n is the number of different trees.

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