Large Permutations

Dartmouth College, Hanover NH

Investigators

Abstract

A permutation is an arrangement of objects in some order: cards in a deck, atoms in a linear polymer, physical or historical events in a chronology. This award supports research into the understanding of permutations from a mathematical perspective, in particular permutations involving large numbers of objects. Powerful new techniques at the interface of combinatorics and probability allow the large-scale study of the behavior of ensembles of permutations. Experience with other large combinatorial objects has led mathematicians to expect them to behave in certain predictable ways. As a result, when such permutations arise in nature and do not have the predicted properties, scientists know to seek a hidden explanation. For example, this involves the study of the significance of the order of genes on a chromosome, and the knowledge of the behavior of ensembles of large random permutations may tell geneticists when to look for a pattern. New discoveries have made it possible to learn much more about the nature of large permutations. In particular, the space of permutations of all sizes can be completed by limit objects that are probability measures on the unit square with uniform marginals. By maximizing a certain entropy integral, one can, in principle, find the limit object that describes nearly all permutations with certain given properties. The proposer and his collaborators will be using both analytic and experimental techniques to exploit this variational principle. One important objective, barely begun, is to plot the landscape of permutations with two fixed pattern densities and determine, for each feasible point, the number and character of the corresponding permutations.