Patterns in Permutations
It is only within that last 15 years that the study of patterns in permutations
has become a recognized field of research. But pattern theory has attracted a lot of
attention because of its beautiful results, intriguing conjectures, and connections
to various disciplines such as combinatorics, algebraic geometry, and computer
science. Because the area is young, it still contains many accessible problems on
which to work.
To explain the basic concepts, we will need some definitions. More details can
be found in the text of Bona [1]. Let σ = a1 a2 . . . ak and τ = b1 b2 . . . bk be two
sequences of distinct integers. Call σ and τ order isomorphic if, for all i and j,
we have ai < aj precisely when bi < bj . So the elements of σ and τ are in the
same relative order. For example σ = 132 and τ = 475 are order isomorphic since
in both the smallest element is first, the largest second, and the one of middle
size at the end. Let Sn be the symmetric group which consists of all permutation
π = a1 an . . . an of the numbers 1, 2, . . . , n. Then we say that σ ∈ Sn contains a
pattern π ∈ Sk if there is a subsequence of σ order isomorphic to τ . To illustrate,
σ = 6432751 contains the pattern σ = 132 because σ is order isomorphic to the
subseqeuce 475 (as well as others).
It turns out that it is often interesting to look at the opposite of pattern containment. Say that σ avoids π if it does not contain π as a pattern. For any
permutation π, let
Avn (π) = {σ ∈ Sn | σ avoids π}.
An interesting fact about avoidance is that for any π ∈ S3 we have
2n
1
# Avn (π) =
n+1 n
where the pound sign denotes cardinality. The quantity on the right-hand side
of this equation is called a Catalan number and these numbers are ubiquitous in
combinatorics.
Note that, by the previous equation, the number of permutations avoiding a
given permutation in S3 does not depend on which permutation is chosen! Call
two permutations π and σ Wilf equivalent, witten π ∼ σ, if # Avn (π) = # Avn (σ)
for all n ≥ 1. This concept is named after Herbert Wilf who was one of the
founders of the study of patterns. So we know that any two permutations in S3
are Wilf equivalent. Some Wilf equivalences are easy to explain. For example, if
π r is the reverse permutation obtained by reading π backwards, then it is easy to
see that π ∼ π r . Other equivalences are not at all obvious. In a paper written
just last year, Sagan and Savage [2] proposed a refinement of the notion of Wilf
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equivalence based on using permutation statistics (which are functions from Sn
into the nonnegative integers). They left it as an open problem to work out the
properties of this new concept and that will be the focus of this REU project.
References
[1] Bóna, Miklos. Combinatorics of permutations. Discrete Mathematics and
its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2004.
[2] Sagan, Bruce E., and Savage, Carla D. Mahonian pairs. Preprint
arXiv:1101.4332.
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