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 $\{1,\dots ,n\}$ avoiding the pattern $\tau$ for all $\tau \in {\mathfrak{S}}_3$ and the Gray codes we obtain have distances 4 or 5.