One important and often difficult problem in the study of restricted
permutations is the enumeration problem: Given a set
$R$ of permutations, enumerate the set $S_n(R)$ consisting of those permutations
in $S_n$ which avoid every element of $R$. The earliest solution to an
instance of this problem seems to be MacMahon's
enumeration of $S_n(123)$. The first explicit solution seems to be
Hammersley's enumeration of $S_n(321)$.
Knuth shows that for any $\sigma \in S_3$, we have
$|S_n(\sigma)| = C_n=\frac{1}{n+1}\binom{2n}{n}$, the $n$th Catalan number.
Other authors considered restricted permutations in the 1970s and early 1980s
(see, for instance, Rogers, Rotem) but the first
systematic study was not undertaken until 1985, when
Simion and Schmidt solved the enumeration problem for every subset of
$S_3$. Recently, there more than 200 papers on this subject.
The idea of this annual conference is due to Michael Atkinson, who announced in first in July of 2001.
Previous conferences:
2003: Conference on Permutation Patterns PP'03, 10-14 February, 2003, University of Otago Dunedin, New Zealand.
The First International Conference of Permutation Patterns was organized by Michael Atkinson, at the University of Otago, in Dunedin, New Zealand, February 10-14, 2003. Keynote speaker was Herb Wilf, from the University of Pennsylvania. The Electronic Journal of Combinatorics devoted a special issue to the conference.
2004: Conference on Permutation patterns PP'04, 5-9 July, 2004, Malaspina University-College, Nanaimo, British Columbia, Canada.
The Second
International Conference of Permutation Patterns is organized by
Julian West, at Malaspina College, at Nanaimo, British Columbia, Canada, July
2-9, 2004. Keynote Speaker is Miklos Bona, from the University of Florida.
Annals of Combinatorics is devoting a special issue to the conference.