 
Summary: Permutations generated by stacks and deques
M. H. Albert
M. D. Atkinson
S. A. Linton
June 23, 2009
Abstract
Lower and upper bounds are given for the the number of permutations
of length n generated by two stacks in series, two stacks in parallel, and
a general deque.
Keywords: Deque, parallel stacks, serial stacks, permutation, enumer
ation, growth rate.
Mathematics Subject Classification: Primary 05A05, 05A16; Sec
ondary 68P05
1 Introduction
In the exercises of section 2.2.1 of the first edition of volume 1 of The Art of
Computer Programming [6] Donald Knuth raised some fascinating questions
about the number of permutations that various data structures could generate.
The most famous of these was about the number of permutations that could be
generated by a stack which he solved completely. Another scenario that he com
pletely solved concerned the number of permutations that could be generated by
