Explanation Exponential generating functions are useful for solving the problems to which ordinary generating functions are not applicable. In most of the problems, it is common that the problem begins for each n with a set of n elements and then it is asked to count the all possible ways of imposing a certain structure upon that given set. Thus for imposing the structure we arrange the elements of that set into blocks. For a permutation, we need to arrange the set in some order, for a subset we arrange the elements into those chosen to belong the subset and those not, for a tree we arrange the elements into the vertices of tree. Ordinary generating functions arise when we have a finite set or countable infinite set of objects S and a weight function w : S → N r , then the ordinary generating function Φ S w ( X ) is defined and we can proceed with calculation, but exponential generating functions are useful when the terms of ( a i ) i = 0 ∞ arise from the counting of permutations. For a set of n elements, there are so many different ways to form a structure. For partitioning, we arrange the n elements into blocks in different fashion. In a permutation it is arranged in a specific manner, the set is of n elements, so for a subset, the set is divided into ( n − a ) and a number of elements. But the exponential generating functions are used to solve a variety of counting problems and the formula can be defined as; F ( x ) = ∑ n f ( n ) x n n ! , where f ( n ) is the number of structure on a set of n elements. Before we work on any significant examples of exponential generating function, we will find out the solution for small structure that can be called trivial structure of a set. Trivial structures have little content, formula and structure may be trivial but they are very useful. Imposing a trivial structure on a set means that the elements are given the structures which they already have, it states that there is one structure on every set. Thus the number of trivial structure on n elements set is f ( n ) = 1 for every n . Therefore, the exponential generating function for a trivial structure is; F ( x ) = ∑ n = 0 ∞ x n n ! = e x This generating function is not a solution for all the trivial structure there are many trivial structure and they all have different solutions

Discrete Mathematics and Its Applications

8th Edition

ISBN: 9781260501759

Author: ROSEN

Publisher: MCG

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Discrete Mathematics and Its Applications

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