ABSTRACT
In this chapter, the authors show how to define and count discrete objects using combinatorial calculus. They aim to understand characteristics of a typical element of the class. The authors also aim to use multivariable generating functions to compute statistical information about combinatorial properties of the objects. They seek to extend the coefficient ring of power series to incorporate more variables to track this additional information. The author focus on moment inequalities and concentration. Parameters that are introduced with the intent of facilitating enumeration are called catalytic parameters, and they are marked by catalytic variables. There are numerous general results on the distribution of the parameter number of components. There are many functional equations, particularly related to lattice walks in bounded regions which can be solved using the kernel method.
