Figure generating function

One of the attractions of Polya s paper is the elegant way that generating functions are introduced and used. In order to summarize the information about the figures and their contents, Polya introduced the "figure generating function" (also frequently called the "figure counting series")... 

ThcorcnL The configuration generating function is obtained by substituting the figure generating function in the cycle index, by which is meant replacing every occurrence of s in the cycle index by f(x ). Thus... 

Suppose we wish to enumerate necklaces of six beads, each of which is either red or green. There are thus two figures, and six boxes (locations) in which to put them. Since we may rotate the necklace or turn it over, the appropriate group is the dihedral group Z)g. Let "content" denote "number of green beads". The figure generating function is then 1 + x. Since... 

A variation of the basic problem with indistinguishable boxes occurs when the mapping from boxes to figures is required to be one-to-one, that is, when we are not allowed to put the same figure in two different boxes. In that case Pdlya showed that the substitution of the figure generating function must be made, not in the cycle index Z(S but in the expression... 

To find the number of trees rooted at an edge we have merely to take the distinguished edge and add a rooted tree at each end. This is a Polya-type problem with two interchangeable boxes, and figure generating function T(x). Polya s Theorem thus gives... 

The light shed by Redfield s paper on the close connection between Polya theory and symmetric function theory is well illustrated by a particularly simple way of looking at Polya s Theorem -- one that shows the way to further developments. Suppose the store of figures consists of n distinct figures, as for example with necklace problems using n kinds of beads. The figure generating function is then... 

In the general problem of this type the figure generating function would be that for the allowable radicals, and the group will be the group of automorphisms of the frame, restricted to those atoms which do not enjoy their full valency within the frame. 

The figure generating function for the two sets are // + F + / and X + y (in an obvious notation), and a complete description of all distinct possibilities is given by the configuration generating function... 

The series (2) of Sec. 3, too, is a generating function the collection of figures comprises the planted trees which are topologically different. The nodes of the rooted trees play the role of the balls in the figure there is only one category of balls, and thus the series depends only on one variable. Figure 1 indicates how the figures (planted trees) of the same content (number of nodes) are combined in the coefficients. 

That is, F(x,y,z) is the generating function of the number of nonequivalent configurations. The solution of our problem consists in expressing the generating function F(x,y,z) in terms of the generating function /(x,y,z) of the collection of figures and the cycle index of the permutation group H. 

We apply the assumption to the special collection of figures whose generating function is... 

More generally the content of a figure will be a vector of nonnegative integers. Polya frequently used vectors of dimension 3. In that case the generating functions will be functions of three variables, and the statement of Polya s Theorem then gives... 

Figure 3. Dimensionless heat generation function fg and removal function r vs. dimensionless temperature 6 at different values of a time-decaying parameter i (3)...

Another reaction performed in the dead-end reactor discussed before, is the allylic amination of 3-phenyl-2-propenyl-carbonic acid methyl ester with morpholine. [30] First and second generation commercially available DAB-dendrimers were functionalized with diphenylphosphine groups (Figure 4.13). Two different membranes were used, the Nadir UF-PA-5 (ultrafiltration) and the Koch MPF-50 (former SELRO) (nanofiltration), which gave retentions of 99.2% and 99.9% respectively for the second generation functionalized dendrimers. 

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