 
Summary: Substitutionclosed pattern classes
M. D. Atkinsona,
, N. Ruskucb
, Rebecca Smithc,1
aDepartment of Computer Science, University of Otago, New Zealand
bSchool of Mathematics and Statistics, University of St Andrews, UK
cDepartment of Mathematics, SUNY Brockport, USA
Abstract
The substitution closure of a pattern class is the class of all permutations ob
tained by repeated substitution. The principal pattern classes (those defined by
a single restriction) whose substitution closure can be defined by a finite number
of restrictions are classified by listing them as a set of explicit families.
Keywords: permutation, pattern class, substitution, basis
2000 MSC: 05A05, 05A15, 05A16
1. Introduction
A pattern class of permutations is a set of permutations closed under taking
subpermutations. We recall that a permutation is a subpermutation of a
permutation if contains a subsequence which is isomorphic to (its terms are
ordered relatively the same as those of ). It is convenient to have a phrase for
the opposite condition: avoids if is not a subpermutation of . Similarly,
