# Seminars & Colloquia Calendar

Discrete Math

## How Prolific is a Random Permutation?

#### Peter Winkler (Dartmouth College)

Location:  Hill 705
Date & time: Monday, 24 February 2020 at 2:00PM - 3:00PM

Abstract:

A "pattern" of length k in a permutation $$P$$ in $$S_n$$ is a permutation in $$S_k$$ determined by choosing $$k$$ elements from {1,2,...,n} and looking at the order of their images under $$P$$. For example, if $$P_3 > P_5$$ then the positions 3 and 5 produce the pattern 21.

$$P$$ is "$$d$$-prolific" if every pattern of length n-d is different; equivalently, the set of patterns of $$P$$ of length $$n-d$$ has size $$n$$ choose $$d$$.

We show that the probability that a uniformly random $$P$$ in $$S_n$$ is $$d$$-prolific tends to $$e^{-d^2-d}$$ as $$n$$ grows.

Joint work with Simon Blackburn and Cheyne Homberger.

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