Generating and enumerating 321-avoiding and skew-merged simple permutations

TL;DR

This paper presents a uniform method to explicitly enumerate simple permutations in the classes of 321-avoiding and skew-merged permutations, providing new structural insights and extending to grid classes of infinite paths.

Contribution

It explicitly enumerates simple permutations in these classes, offering the first structural enumeration for skew-merged permutations and extending the method to broader classes.

Findings

Explicit enumeration of simple 321-avoiding permutations

First structural enumeration of skew-merged permutations

Extension of the method to grid classes of infinite paths

Abstract

The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards from the enumeration of the class, but the simple permutations had not been enumerated explicitly. In particular, the enumeration of the simple skew-merged permutations leads to the first truly structural enumeration of this class as a whole. The extension of this method to a wider collection of classes namely grid classes of infinite paths is discussed.