# Wilf's conjecture in fixed multiplicity

## TL;DR

This paper presents an algorithm based on polyhedral geometry to verify Wilf's conjecture for numerical semigroups with fixed multiplicity, successfully proving it for all cases where the multiplicity is at most 18.

## Contribution

It introduces a novel algorithm combining polyhedral techniques and a combinatorial game to verify Wilf's conjecture for fixed multiplicities, extending the known cases.

## Key findings

- Wilf's conjecture holds for all numerical semigroups with multiplicity up to 18.
- Developed a parallelizable algorithm for face enumeration of polyhedral cones.
- Introduced a combinatorial game method for conjecture verification.

## Abstract

We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity $m$, and use it to prove Wilf's conjecture holds whenever $m \le 18$. 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.