Permutational isomers by content

2 Permutational Isomers by content, in particular by racemic content 

As in enumeration of multigraphs by weight or content (Example 1.35), the total number of permutational isomers, or of achiral or chiral ones, as shown above, can be refined substantially by evaluating the number of permutational isomers or distributions d with specified content 

17 Definition (Racemic) A distribution S as well as its content con(S) are called racemic if S takes the value y eY as often as the enantiomorphic value y e Yc).  

The set of distributions with racemic content will be denoted by 

In the previous section we obtained sizes of combinatorial bbraries, e.g. numbers of eunidations of xanthene-tetracarboxylic acid chloride, euid also sizes of sublibraries contauiing achiral or chiral amidations only. Some of these numbers can be obtained in one step, e.g. the number of permutational isomers with racemic content. The answer to such problems will be a generating Junction, a polynomial in the indetermi-nates y Y that has the desired numbers as coefficients of monomial summands. Generating firnctions for the numbers of simple graphs with given number of bonds were obtained above. The number of permutational isomers of content con S) will be the coefficient of the monomial summand 

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The crucial point is that this generating function for the numbers of permutational isomers by content and with respect to R suffices when we use the notion of racemic content. We demonstrate this using the amidation of xanthene-tetracarboxylic acid chloride, using three types of ligands, a pair of enantiomorphic ones, x and y and the self-enantiomorphic glycine, indicated by z. 

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