Decomposing simple permutations, with enumerative consequences

TL;DR

This paper proves that long simple permutations contain two long almost disjoint simple subsequences, leading to new enumeration results for permutations with restricted patterns, such as those with limited 132 patterns, with algebraic generating functions.

Contribution

It establishes a structural property of simple permutations that enables new enumeration results for pattern-restricted permutations.

Findings

Long simple permutations contain two long almost disjoint simple subsequences.

Permutations with bounded copies of 132 have algebraic generating functions.

Abstract

We prove that every sufficiently long simple permutation contains two long almost disjoint simple subsequences. This result has applications to the enumeration of restricted permutations. For example, it immediately implies a result of Bona and (independently) Mansour and Vainshtein that for any r, the number of permutations with at most r copies of 132 has an algebraic generating function.