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

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