Year: 2025
Type: article
Abstract: We study the process of bootstrap percolation on n × n permutation matrices, inspired by Shapiro and Stephens' article (SS): "Bootstrap percolation, the Schröder numbers, and the n-kings problem" (1991).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 (SS), that the number of full n × n permutations is the (n -1) st large Schröder number.Finally, after rigorously justifying a heuristic argument in (SS), we find a new formula for the number of n × n "no growth" permutation matrices, and hence a new solution to the well-known n-kings problem.
Institutions: Skidmore College, Wentworth Institute of Technology
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4416454871
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