Permutations with fixed pattern densities
Combinatorics
2015-09-01 v2 Probability
Abstract
We study scaling limits of random permutations ("permutons") constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed \hbox{12} density, with fixed \hbox{12} and \hbox{123} densities, with fixed \hbox{12} density and the sum of \hbox{123} and \hbox{213} densities, and with fixed \hbox{123} and \hbox{321} densities. In the last case we explore a particular phase transition. To obtain our results, we also provide a description of permutons using a dynamic construction.
Cite
@article{arxiv.1506.02340,
title = {Permutations with fixed pattern densities},
author = {Richard Kenyon and Daniel Kral and Charles Radin and Peter Winkler},
journal= {arXiv preprint arXiv:1506.02340},
year = {2015}
}