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A wide variety of problems in combinatorics and discrete optimization depend on counting the set  S  of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour through numerous problems of this type. In particular, we consider families of such sets  S tdepending on one or more integer parameters  t , and analyze the behavior of the function. In the examples that we investigate, this function exhibits surprising polynomial-like behavior. We end with two broad theorems detailing settings where this polynomial-like behavior must hold. The plethora of examples illustrates the framework in which this behavior occurs and also gives an intuition for many of the proofs, helping us create a toolbox for counting problems like these.

A Plethora of Polynomials: A Toolbox for Counting Problems