Pattern Avoiding Permutations as Walks

TL;DR

This paper develops a graph-based method to estimate the Stanley-Wilf limit for pattern 1324 by encoding permutations as walks, providing improved lower bounds under a conjecture.

Contribution

It introduces a novel graph encoding approach for permutations and derives new lower bounds on the Stanley-Wilf limit for pattern 1324.

Findings

Lower bound of 10.418 on the Stanley-Wilf limit for pattern 1324

Graph encoding relates permutation growth to spectral radius of adjacency matrices

Conditional bounds depend on a natural conjecture

Abstract

The Stanley-Wilf limit of the pattern 1324 is known to lie between 10.271 and 13.5. We obtain lower bounds on this limit by encoding permutations as walks in directed graphs: building a permutation by successive insertion of maxima corresponds to traversing edges, and the growth rate of walks equals the spectral radius of the adjacency matrix. For 1324, this graph is too large for direct computation, so we pass to a quotient graph with weighted edges. Conditional on a natural conjecture, this yields a lower bound of 10.418.