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Pattern avoiding permutations and Brownian excursion

Mathematical Physics & Probability

Speaker: Erik Slivken, University of Washington
Location: 1147 MSB
Start time: Wed, Feb 26 2014, 2:10PM

Permutations of size n that avoid a given pattern from S_3 can be counted by the nth Catalan number. Dyck paths of length 2n can also be counted by the nth Catalan number. Naturally there exists bijections between the two sets. Remarkably, given the right choice of bijection, both of these random objects converge (in some sense) to the same thing, Brownian excursion. This connection to Brownian excursion helps answer all sorts of questions about the permutations. For example, we show a striking connection between the density of fixed points of a 231-avoiding permutation and the height of the corresponding Brownian excursion. This is joint work with Christopher Hoffman and Douglas Rizzolo.