Combinatorial Gray codes for classes of pattern avoiding permutations

TL;DR

This paper develops combinatorial Gray codes for efficiently generating pattern avoiding permutations across various families, enabling exhaustive enumeration with minimal changes, which is useful for applications requiring ordered permutation generation.

Contribution

It introduces new algorithms and Gray codes for pattern avoiding permutations, covering multiple well-known combinatorial families and providing efficient exhaustive generation methods.

Findings

Gray codes with distances 4 and 5 for pattern avoiding permutations

Algorithms for families counted by Catalan, Schröder, Pell, Fibonacci, and binomial coefficients

Exhaustive generation of permutations with specific pattern constraints

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, Schr\"oder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray codes for $\s_n(\tau)$ for all $\tau\in \s_3$ and the obtained Gray codes have distances 4 and 5.