Combinatorial generation via permutation languages. II. Lattice congruences

TL;DR

This paper explores lattice congruences of the weak order on permutations, analyzing the structure of associated graphs and polytopes, proving Hamiltonicity, and connecting these to pattern-avoiding permutations.

Contribution

It introduces a framework for generating combinatorial objects via permutations, characterizes properties of quotient graphs, and links lattice congruences to pattern avoidance.

Findings

All quotient graphs have a Hamilton path, found by a greedy algorithm.

Characterized when these graphs are vertex-transitive or regular.

Provided counting results and degree bounds for the graphs.

Abstract

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406-420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees. Moreover, we investigate the relation between lattice congruences of the weak order and pattern-avoiding permutations.