On the asymptotic enumeration and limit shapes of monotone grid classes of permutations

TL;DR

This paper develops a method to asymptotically count permutations in monotone grid classes, computes their asymptotic numbers, and determines their limit shapes, advancing understanding of their combinatorial structure.

Contribution

It introduces a novel procedure for asymptotic enumeration and limit shape analysis of monotone grid classes of permutations, including connected classes.

Findings

Asymptotic enumeration formulas for monotone grid classes.

Explicit limit shapes for connected monotone grid classes.

Analysis of the distribution of points within typical large permutations.

Abstract

We exhibit a procedure to asymptotically enumerate monotone grid classes of permutations. This is then applied to compute the asymptotic number of permutations in any connected one-corner class. Our strategy consists of enumerating the gridded permutations, finding the asymptotic distribution of points between the cells in a typical large gridded permutation, and analysing in detail the ways in which a typical permutation can be gridded. We also determine the limit shape of any connected monotone grid class.