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Mean and Insensitive. Random Permutations.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
Set gn(z)=∑k=0m(n)pn(k)zk. In other words, gn(z) is a generating function satisfying gn(1)=1. Such generating functions are sometimes called probability generating functions.
Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
A generating function is a way of encoding a sequence {an}n≥0 as a single algebraic object—an infinite power series—which can then be added, multiplied, and otherwise combined with other sequences. Generating functions come in ordinary and exponential forms (among others), and each will be useful for different situations.
Combinatorics
Published in B. V. Senthil Kumar, Hemen Dutta, Discrete Mathematical Structures, 2019
B. V. Senthil Kumar, Hemen Dutta
In this section, we will show how recurrence relations can be solved using the powerful generating function method. Generating function is an important tool in discrete mathematics, and its use is by no means confined to the solution of recurrence relations.
Approximate Methods for Inverting Generating Functions from the Pál-Bell Equations for Low Source Problems
Published in Nuclear Science and Engineering, 2019
In general, the generating function is not known explicitly but is the solution of a differential equation. This means that one must solve simultaneously the time-dependent equations for . If space and energy are also included this can lead to a very lengthy numerical calculation and it is useful to see if any further approximations can be made to simplify the numerical work. We have already suggested the Gamma pdf above. It is at this point that we return to some work of Bell10 who shows that for a slowly varying function of we may write