Speaker: Christopher Hoffman
Title: The shape of a random pattern-avoiding permutation
Abstract:
A permutation that avoids the pattern 4321 has a longest decreasing sequence of
length 3. We fix n, choose $\sigma$ a 4321-avoiding permutation uniformly at random
and plot the points of the form $(i/n,\sigma(i)/n)$ for $1 \leq i \leq n$. Looking at this plot
it is clear that the indices 1 through n can be partitioned into three sets. By linear
interpolation from these three sets we can generate three functions. We show that the
scaling limit of this measure on triples of functions is given by the eigenvalues of a
ensemble of random matrices. We also discuss the scaling limits of other patterns.