On Ternary Inclusion-Exclusion Polynomials

Abstract

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials $\Phi_{pqr}$, where $p<q<r$ are odd primes. Our main result is that the set of coefficients of $\Phi_{pqr}$ is simply a string of consecutive integers which depends only on the residue class of $r$ modulo $pq$.