Combinatorial generation via permutation languages. II. Lattice congruences

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 vertextransitive 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.

Combinatorial Gray codes-an updated survey

“…• The hypercube, permutahedron and associahedron arise as cover graphs of quotients of lattice congruences of the symmetric group S n [Rea12], and the corresponding polytopes are known as quotientopes [PS19,PPR20]. This is a family of 2 2 n(1+o(1)) many flip graphs, and there is a simple greedy algorithm for computing a Hamilton path in each of them [HM21], which yields a unified description of several known Gray codes for bitstrings, permutations, binary trees, set partitions, rectangulations etc. [HHMW21].…”

“…This generation approach therefore generalizes several known Gray codes, such as the BRGC (for X = {231, 132}), the SJT algorithm (for X = ∅ minimal jumps are adjacent transpositions), the Gray code for binary trees by rotations due to Lucas, Roelants van Baronaigien and Ruskey [LRvBR93] (for X = {231}), and the Gray code for set partitions by element exchanges described by Kaye [Kay76] (for X = {231}). It also yields Gray codes for many different classes of rectangulations [MM21], and it produces Hamilton paths and cycles on large classes of polytopes [HM21,CMM21]. The reason why minimal jumps are natural becomes clear when considering the inversion vector of the permutations.…”

Combinatorial generation via permutation languages. IV. Elimination trees

“…The reason we refer to [HM21] here instead of [HHMW21] is that the notion of zigzag language given in [HHMW21] omits condition (z2) before. However, for the present paper we need the more general definition with condition (z2) introduced in [HM21] to be able to handle elimination forests for disconnected graphs G. Based on Theorem 7, for any zigzag language L n ⊆ S n , we write J(L n ) for the sequence of permutations from L n generated by Algorithm J with initial permutation π 0 = id n .…”

“…It was shown in [HM21] that the sequence J(L n ) can be described recursively as follows. For any π ∈ L n−1 we let #" c (π) be the sequence of all c i (π) ∈ L n for i = 1, 2, .…”

“…It turns out that this algorithm can be implemented efficiently for many combinatorial classes of objects, and subsumes several previously studied Gray codes. In a series of recent papers, this framework was applied to a plethora of combinatorial objects such as pattern-avoiding permutations [HHMW21], lattice quotients of the weak order on permutations [HM21], and rectangulations [MM21]. In this work, we extend the reach of this framework and make it applicable to the efficient generation of structures on graphs, specifically of elimination forests, which is a step forward in exploring the generality of this framework.…”