# Permuted packings and prolific permutations

The event is taking part on the Tuesday, Nov 28th 2017 at 15.30
Theme/s: Combinatorics, Pure and Applied Colloquia, Pure Maths
Location of Event: Alan Turing Room 306
This event is a: Public Seminar

Abstract: Given a toast rack with $n$ slots, in how many ways can the slices be removed so that no two consecutive slices are removed from adjacent slots?

A permutation of length $n$ is $k$-prolific if each of the $(n-k)$-subsets of the entries in its one-line notation forms a distinct pattern.
This talk will address the following questions: How can $k$-prolific permutations be characterised? How small can a $k$-prolific permutation be?
What proportion of large permutations are $k$-prolific? And, what has this to do with the toast rack? In answering them, use is made of a nice bijection between $k$-prolific permutations and certain packings of diamond-shaped tiles, which we call permuted packings.

This talk is based on joint work with Cheyne Homberger and Bridget Tenner and on subsequent work by Cheyne, Simon Blackburn and Peter Winkler.