Bootstrap Percolation, Indecomposable Permutations, and the n-Kings problem

TL;DR

This paper investigates bootstrap percolation on permutation matrices, explores their connection to indecomposable permutations, and provides new formulas and solutions for related combinatorial problems like the n-kings problem.

Contribution

It introduces a novel analysis of bootstrap percolation on permutation matrices and establishes new formulas for permutation counts, including a solution to the n-kings problem.

Findings

Number of full indecomposable permutations is half of all full permutations.

Number of full permutations equals the (n-1)st large Schroeder number.

Derived a new formula for the count of no-growth permutations.

Abstract

We study the process of bootstrap percolation on n x n permutation matrices, inspired by the work of Shapiro and Stephens [5]. In this percolation model, cells mutate (from 0 to 1) if at least two of their cardinal neighbors contain a 1, and thereafter remain unchanged; the process continues until no further mutations are possible. After carefully analyzing this process, we consider how it interacts with the notion of (in)decomposable permutations. We prove that the number of indecomposable permutations whose matrices "fill up'' to contain all 1's (or are "full") is half of the total number of full permutations. This leads to a new proof of a key result in [5], that the number of full n x n permutations is the (n-1)st large Schroeder number. Finally, after rigorously justifying a heuristic argument in [5], we find a new formula for the number of n x n "no growth" permutations, and hence a new solution to the well-known n-kings problem.