A Plethora of Polynomials: A Toolbox for Counting Problems
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-t depending on one or more integer parameters t, and analyze the behavior of the function f(t) = vertical bar S-t vertical bar. 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.
Bogart, Tristram, and Kevin Woods. 2022. "A Plethora of Polynomials: A Toolbox for Counting Problems." The American Mathematical Monthly 129(3): 203-222.
Taylor & Francis
American Mathematical Monthly
Factorization length distribution, Numerical semigroups, Integer