Exhaustive generation for permutations avoiding a (colored) regular sets of patterns

TL;DR

This paper introduces a general framework for exhaustively generating permutations that avoid certain regular and colored regular patterns, expanding the toolkit for pattern-avoiding permutation classes with efficient algorithms.

Contribution

It defines regular and colored regular pattern sets, proves their regularity, and develops succession-function-based algorithms for their exhaustive generation.

Findings

Algorithms for generating pattern-avoiding permutations are derived.

Several classes have counting sequences like Pell, Fibonacci, and Catalan.

The framework encompasses classes with variable length forbidden patterns.

Abstract

Despite the fact that the field of pattern avoiding permutations has been skyrocketing over the last two decades, there are very few exhaustive generating algorithms for such classes of permutations. In this paper we introduce the notions of regular and colored regular set of forbidden patterns, which are particular cases of right-justified sets of forbidden patterns. We show the (colored) regularity of several sets of forbidden patterns (some of them involving variable length patterns) and we derive a general framework for the efficient generation of permutations avoiding them. The obtained generating algorithms are based on succession functions, a notion which is a byproduct of the ECO method introduced in the context of enumeration and random generation of combinatorial objects by Barcucci et al. in 1999, and developed later by Bacchelli et al. in 2004, for instance. For some classes of permutations falling under our general framework, the corresponding counting sequences are classical in combinatorics, such as Pell, Fibonacci, Catalan, Schr\"oder and binomial transform of Padovan sequence.