On Ternary Inclusion-Exclusion Polynomials

TL;DR

This paper introduces inclusion-exclusion polynomials from a combinatorial perspective, focusing on ternary cases like cyclotomic polynomials, and reveals that their coefficients form a consecutive integer string determined by prime residue classes.

Contribution

It extends the study of cyclotomic polynomials by defining inclusion-exclusion polynomials and characterizes the coefficient structure of ternary cases based on prime residue classes.

Findings

Coefficients form a string of consecutive integers

Coefficient structure depends only on residue class of r modulo pq

Provides a detailed analysis of 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 $pq$.