Counting by weight

The coefficient 3 of its monomial summand Sy yl indicates that there are exactly three unlabeled simple graphs of content (3,3), which means three bonds of multiplicity 0 (non-bonds) and three bonds of multiplicity 1 (single bonds). We shall show how this generating function can be obtained. 

We denote the set of orbits of length i of g on X by ( ), X, recalling that this number is the number of i-cycles of the permutation g induced by e G on the setX, 

According to Polya, the desired number of symmetry classes y of mappings y y of weight c = (cq,. .., c ) is the coefficient of the monomial Oiem the 

We call the resulting polynomial the groug reduction function. 

32 Exercise Evaluate the cycle index C(G,X) of the symmetry group of naphthalene (see Example 1.7) and replace the i-th indeterminate Zf by the polynomial yg + y. Derive the corresponding numbers of symmetry classes of the symmetry group on the set of mappings = 2 by weight. 

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SO that unlabeled m-multigraphs can be counted. Moreover, symmetry classes can be counted by weight, and the generating function can be obtained from the cycle index of qX. This approach was used above to construct unlabeled m-multigraphs and to obtain a method for generating unlabeled m-multigraphs uniformly at random. This needs to be generalized because we now have to take chirality into account. 

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