 
Summary: Counting (3 + 1)avoiding permutations
M. D. Atkinsona,
, Bruce E. Saganb
, Vincent Vatterc
aDepartment of Computer Science, University of Otago, Dunedin, New Zealand
bDepartment of Mathematics, Michigan State University, East Lansing, USA
cDepartment of Mathematics, University of Florida, Gainesville, USA
Abstract
A poset is (3 + 1)free if it contains no induced subposet isomorphic to the disjoint union of a 3
element chain and a 1element chain. These posets are of interest because of their connection with
interval orders and their appearance in the (3 + 1)free Conjecture of Stanley and Stembridge. The
dimension 2 posets P are exactly the ones which have an associated permutation where i j in
P if and only if i < j as integers and i comes before j in the oneline notation of . So we say that
a permutation is (3 + 1)free or (3 + 1)avoiding if its poset is (3 + 1)free. This is equivalent to
avoiding the permutations 2341 and 4123 in the language of pattern avoidance. We give a complete
structural characterization of such permutations. This permits us to find their generating function.
Keywords: algebraic generating function, permutation class, restricted permutation, simple
permutation, substitution decomposition, (3 + 1)free
2000 MSC: 05A05, 05A15
1. Introduction
