The enumeration of simple permutations

TL;DR

This paper explores the enumeration of simple permutations, establishing a relationship with all permutations, analyzing their generating functions, asymptotic behavior, and congruence properties.

Contribution

It provides a new relationship between generating functions of simple and all permutations, proves non-P-recursiveness, and offers asymptotic and congruence results.

Findings

The generating function for simple permutations relates straightforwardly to that of all permutations.

Coefficients of this generating function are not P-recursive.

Asymptotic expansion and congruence properties of these coefficients are established.

Abstract

A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coefficients of this series are not P-recursive, an asymptotic expansion for these coefficients, and a number of congruence results.