Wilf’s conjecture in fixed multiplicity

Abstract

We give an algorithm to determine whether Wilf’s conjecture holds for all numerical semigroups with a given multiplicity [Formula: see text], and use it to prove Wilf’s conjecture holds whenever [Formula: see text]. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf’s conjecture via a combinatorially flavored game played on the elements of a certain finite poset.