 
Summary: Bull. London Math. Soc. Page 1 of 12 Ce2011 London Mathematical Society
doi:10.1112/blms/bdr022
Subclasses of the separable permutations
Michael H. Albert, M. D. Atkinson and Vincent Vatter
Abstract
The separable permutations are those that can be obtained from the trivial permutation by two
operations called direct sum and skew sum. This class of permutations contains the class of stack
sortable permutations, Av(231), which are enumerated by the Catalan numbers. We prove that
all subclasses of the separable permutations which do not contain Av(231) or a symmetry of this
class have rational generating functions. Our principal tools include partial wellorder (the lack
of an infinite antichain), atomicity (the joint embedding property), and the theory of strongly
rational permutation classes which is introduced here for the first time.
1. Introduction
The separable permutations are those that can be built from the trivial permutation 1 by
repeatedly applying two operations, known as direct sum (or simply, sum) and skew sum (or
simply, skew) which are defined, respectively, on permutations of length m and of length
n by
( )(i) =
(i) if 1 i m,
(i  m) + m if m + 1 i m + n,
