Abstract
We define the operation of composing two hereditary classes of permutations using the standard composition of permutations as functions and we explore properties and structure of permutation classes considering this operation. We mostly concern ourselves with the problem of whether permutation classes can be composed from their proper subclasses. We provide examples of classes which can be composed from two proper subclasses, classes which can be composed from three but not from two proper subclasses and classes which cannot be composed from any finite number of proper subclasses.
Highlights
Permutations of numbers or other finite sets are a very deeply and frequently studied combinatorial and algebraic object
There are two main structures on permutations investigated in modern mathematics: groups, closed under the composition operator, and hereditary pattern-avoiding classes, closed under the relation of containment
This paper is one of several texts exploring the relation between the two notions by applying the composition operator to permutation classes
Summary
Permutations of numbers or other finite sets are a very deeply and frequently studied combinatorial and algebraic object. Atkinson and Stitt [5, Section 6.4] introduce the pop-stack, a sorting machine which sorts precisely the layered permutations (see Section 4.2 for a definition), and consider the class of permutations which can be sorted by two pop-stacks in genuine series, i.e. connected by a queue. This turns out to be the class of permutations that can be written as a composition of two layered permutations.
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