# A new face iterator for polyhedra and more general finite locally branched lattices

## TL;DR

This paper introduces a memory-efficient depth-first algorithm for iterating over elements of finite locally branched lattices, applicable to polyhedral structures and generalizations, with practical speed improvements and applications to Wilf's conjecture.

## Contribution

The paper presents a novel, fast, memory-efficient algorithm for traversing finite locally branched lattices, extending its application to complex geometric and algebraic structures.

## Key findings

- Algorithm is significantly faster than previous methods.
- Successfully applied to verify Wilf's conjecture for all numerical semigroups of multiplicity 19.
- Effective in analyzing face lattices of polyhedra and related structures.

## Abstract

We discuss a new memory-efficient depth-first algorithm and its implementation that iterates over all elements of a finite locally branched lattice. This algorithm can be applied to face lattices of polyhedra and to various generalizations such as finite polyhedral complexes and subdivisions of manifolds, extended tight spans and closed sets of matroids. Its practical implementation is very fast compared to state-of-the-art implementations of previously considered algorithms. Based on recent work of Bruns, Garc\'ia-S\'anchez, O'Neill and Wilburne, we apply this algorithm to prove Wilf's conjecture for all numerical semigroups of multiplicity 19 by iterating through the faces of the Kunz cone and identifying the possible bad faces and then checking that these do not yield counterexamples to Wilf's conjecture.