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.We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the 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 certain 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 the equivalence classes of 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. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.
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...
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.
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.
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