# Statistics in Random Restricted Permutations

United States Naval Academy, Annapolis, MD

## Abstract:

Identifying trends within two-dimensional data is a common challenge across the
sciences, and the theory of permutation patterns adds new tools to this problem.
One permutation is said to occur as a pattern in a larger one if we can find
entries in the larger permutation which are in the same relative order as those
of the smaller. By translating sets of points on a plane to permutations, we
can use the language of permutations to describe and explore patterns. Pattern
occurrences translate to topological invariants of a dataset, the statistics of
which have only recently been studied.

In this talk we investigate the following question: How does the absence of one
pattern affect the number of occurrences of another? This has led to several
interesting and surprising identities, concerning both individual patterns and
the number of patterns with the same distribution across a set of permutations.
We start by exploring the notion of pattern-avoiding sets of permutations,
before analyzing the number of small patterns in pattern-avoiding permutations
and classifying pattern occurrence identities within the separable permutations.