Combinatorial generation via permutation languages

Abstract

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an n-element set by adjacent transpositions; the binary reflected Gray code to generate all n-bit strings by flipping a single bit in each step; the Gray code for generating all n-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an n-element ground set by element exchanges due to Kaye.The first main application of our framework are permutation patterns, yielding new Gray codes for different pattern-avoiding permutations, such as vexillary, skew-merged, separable, Baxter and twisted Baxter permutations, 2-stack sortable permutations, geometric grid classes, and many others. We also obtain new Gray codes for many combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to different restrictions.The second main application of our framework are lattice congruences of the weak order on the symmetric group Sn. Recently, Pilaud and Santos realized all those lattice congruences as (n − 1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. 2 COMBINATORIAL GENERATION VIA PERMUTATION LANGUAGES. I. FUNDAMENTALS for specific classes of objects, and many of these algorithms are covered in depth in the most recent volume of Knuth's seminal series 'The Art of Computer Programming' [Knu11].1.1. Overview of our results. The main contribution of this paper is a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, which provides a unified view on many known results and allows us to prove many new ones. The basic idea is to encode a particular set of objects as a set of permutations L n ⊆ S n , where S n denotes all permutations of [n] := {1, 2, . . . , n}, and to use a simple greedy algorithm to generate those permutations by cyclic rotations of substrings, an operation we call a jump. This works under very mild assumptions on the set L n , and allows us to generate more than double-exponentially (in n) many distinct sets L n . Moreover, the jump orderings obtained from our algorithm translate into listings of combinatorial objects where consecutive objects differ by small changes, i.e., we obtain Gray codes [Sav97], and...

Acyclic reorientation lattices and their lattice quotients

“…In connection to the simpliciality of the quotient fans (or equivalently to the simplicity of the quotientopes) defined below, it would be interesting to understand which quotients of AR D have a regular cover graph (meaning all vertices have the same degree). For instance, when D is a tournament, H. Hoang and T. Mütze proved in [HM19] that the cover graph of AR D /≡ is regular if and only if the generators (as an upper ideal of the subrope order) of the complement of I ≡ are all of the form (u, v, , ∅) or (u, v, ∅, ). We hope that the rope interpretation of the congruences of AR D will help to extend this result for arbitrary skeletal directed acyclic graphs Problem 41.…”

“…Another classical result, proved in [Luc87,HN99], states that the graph of the associahedron admits a Hamiltonian cycle. It was proved recently in [HM19] that the graph of any lattice quotient of the weak order actually admits a Hamiltonian path (the question of the existence of a Hamiltonian cycle remains open in general). The approach of [HM19] being largely based on noncrossing arc diagrams, it motivates the following question, which has been positively answered by computer experiments on all lattice congruences of the acyclic reorientation lattices of all skeletal directed acyclic graphs up to 5 vertices.…”

A Pivot Gray Code Listing for the Spanning Trees of the Fan Graph

“…The work by Williams [23] notes that some very well-known combinatorial listings can be constructed greedily, including the binary reflected Gray code (BRGC) for binary strings, the plain change order for permutations, and the lexicographically smallest de Bruijn sequence. Recently, a very powerful greedy algorithm on permutations (known as Algorithm J, where J stands for "jump") generalizes many known combinatorial Gray code listings including many related to permutation patterns, rectangulations, and elimination trees [10,11,18]. However, no greedy algorithm was previously known to list the spanning trees of an arbitrary graph.…”

“…Thus, answering this third question would satisfy only the first part of Research Question #1. However, in many cases, an underlying pattern can be found in a greedy listing which can result in space efficient algorithms [10,23].…”

Ranking Bracelets in Polynomial Time

“…A variety of such algorithms are assembled in the fourth volume of the prominent series "The art of computer programming" by D. Knuth [10]. Nevertheless, this research direction remains very active [8].…”