More restrictive Gray codes for some classes of pattern avoiding permutations

Baril, Jean-Luc

doi:10.1016/j.ipl.2009.03.025

Cited by 5 publications

(5 citation statements)

References 23 publications

(25 reference statements)

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“…For instance, Knuth [Knu98] first proved that all 123-avoiding and 132-avoiding permutations are counted by the Catalan numbers (see also [CK08]). With regards to counting and exhaustive generation, a few tree-based algorithms for pattern-avoiding permutations have been proposed [Eli07,DFMV08,Bar08,Bar09]. Pattern-avoidance has also been studied extensively from an algorithmic point of view.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial generation via permutation languages

Hartung

Hoang

Mütze

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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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...

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Section: Introductionmentioning

confidence: 99%

Combinatorial generation via permutation languages

Hartung

Hoang

Mütze

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In [8], Dukes et al gave Gray codes for large families of pattern avoiding permutations including many fundamental classes. Baril improved their results [1]. Their proofs are based on ECO method [2] [4].…”

Section: Introductionmentioning

confidence: 93%

“…The most fundamental cases are Gray codes for the permutations of length n avoiding a single pattern of length three. In particular, Baril constructed Gray codes for S n (p) for p ∈ S 3 , where two consecutive permutations differ by at most three positions and his results are optimal for odd n [1]. Next, we should consider the permutations of length n avoiding two patterns of length three.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian cycles for the square of the augmentation graphs and Gray codes for restricted permutations and ascent sequences

Tomie¹

2016

Preprint

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In this paper, we construct a listing for the vertices of the augmentation graph of given size, and as a consequence, we obtain a Hamiltonian cycle for the square of the augmentation graph of given size. As applications, we have a Gray code for the 132-312 avoiding permutations of given length such that two successive permutations differ by at most 2 adjacent transpositions. Also we obtain Gray codes of strong distance 2 for the 001 avoiding ascent sequences and the 010 avoiding ascent sequences of given length.

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“…We call a pair of integers (i, j) an arc of a set partition π if i and j occur in the same block and j is the least element of the block greater than i; the first coordinate i of an arc (i, j) is defined to be an opener and the second coordinate j is defined to be a closer of the set partition π . For example, π = ({1, 2, 4}, {3, 5}, {6}) is a set partition of [6] with three blocks B 1 = {1, 2, 4}, B 2 = {3, 5} and B 3 = {6}, and the set of arcs is {(1, 2), (2,4), (3,5)}, {1, 2, 3} are the openers and {2, 4, 5} the closers of π . So it is possible for an integer to be at the same time both an opener and a closer.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%