Combinatorial Gray codes for classes of pattern avoiding permutations

Abstract

The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, large Schröder, Pell, even-index Fibonacci numbers and the central binomial coefficients. We thus provide Gray codes for the set of all permutations of {1, . . . , n} avoiding the pattern τ for all τ ∈ S 3 and the Gray codes we obtain have distances 4 or 5.

More restrictive Gray codes for some classes of pattern avoiding permutations

“…This construction is a refinement of that given in [11] and, in some cases, yields codes that are optimal. The construction is based on the method ECO [4].…”

“…We will denote this permutation γ by φ(i, σ ) and we will say that γ is a successor (or son) of σ . The active sites of a permutation σ ∈ S n (T ) are right-justified (see [11]) if all sites to the right of the leftmost active site are also active. We denote by χ T (i, σ ) the number of active sites of φ(i, σ ).…”

“…We denote by χ T (i, σ ) the number of active sites of φ(i, σ ). A set of patterns T is called regular [11] if for any n 1 and γ ∈ S n (T ), -γ has at least two active sites, and the active sites of γ are right-justified, -there exist σ ∈ S n−1 (T ) and an integer i, 1 i n, such that γ = φ(i, σ ), -χ T (i, σ ) does not depend on σ but only on the number k of active sites of σ ; in this case, we will denote…”

“…(n − 1)n and the call gen_down (1,2) produces the Gray code S n (T ) (the procedure gen_ down(size, k) consists in the statements of gen_up(size, k) in reverse order). This algorithm is inspired from the one in [11]. Only the order of the statements are modified according to the two sequences L and L .…”

“…In a recent paper, Dukes et al [11] give Gray codes and generating algorithms for classes of pattern avoiding permutations enumerated by Catalan, Schröder, Pell, even index Fibonacci numbers and the central binomial coefficients: two consecutive elements in their codes differ in at most four or five positions. Inspired by their article, we give here Gray codes for Catalan permutations and for 0020-0190/$ -see front matter © 2009 Elsevier B.V. All rights reserved.…”