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Pattern Avoiding Permutations as Walks

Atli Fannar Franklín

TL;DR

Problem: determine lower bounds for the Stanley-Wilf limit of the pattern 1324. Approach: encode pattern-avoiding permutations as walks built by inserting new maxima; analyze growth via the adjacency-matrix spectral radius and use a quotient graph with weighted edges, conditional on a conjecture that weights bound unweighted walks. Contributions: obtains a conditional lower bound of $L(1324) ≥ 10.418$ and demonstrates the method on smaller patterns, together with a framework using a binary-tree decomposition and stationary-distribution-inspired weights to enable computations. Significance: provides a scalable, graph-theoretic approach to hard pattern-avoiding growth rates and highlights avenues for tightening bounds with more computation and refined partitions.

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.

Pattern Avoiding Permutations as Walks

TL;DR

Problem: determine lower bounds for the Stanley-Wilf limit of the pattern 1324. Approach: encode pattern-avoiding permutations as walks built by inserting new maxima; analyze growth via the adjacency-matrix spectral radius and use a quotient graph with weighted edges, conditional on a conjecture that weights bound unweighted walks. Contributions: obtains a conditional lower bound of and demonstrates the method on smaller patterns, together with a framework using a binary-tree decomposition and stationary-distribution-inspired weights to enable computations. Significance: provides a scalable, graph-theoretic approach to hard pattern-avoiding growth rates and highlights avenues for tightening bounds with more computation and refined partitions.

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.
Paper Structure (6 sections, 7 theorems, 9 equations, 12 figures)

This paper contains 6 sections, 7 theorems, 9 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Motivating examples
  3. 1324-avoiding Graph
  4. Open questions
  5. Acknowledgements
  6. Appendix

Key Result

Theorem 1

Let $A$ be an aperiodic irreducible non-negative $N \times N$ matrix with spectral radius $r$ (largest absolute value among all eigenvalues). Then $r\in\mathbb{R}_{>0}$ is an eigenvalue of $A$ which we call the Perron-Frobenius eigenvalue. This eigenvalue is simple and both the left and right eigens

Figures (12)

  • Figure 1: Construction of $\mathop{\mathrm{\operatorname{Av}}}\nolimits(213)$ graph with cutoff $N = 3$
  • Figure 2: Graph of $\mathop{\mathrm{\operatorname{Av}}}\nolimits(2134)$ with cutoff $N = 3$, version one
  • Figure 3: Filtered $\mathop{\mathrm{\operatorname{Av}}}\nolimits(2134)$, cutoff $N = 4$, version two
  • Figure 4: Quotient graph on $B(n, r)$ for cutoff $N = 5$
  • Figure 5: Stanley-Wilf estimate of $2134$ as a function of cutoff
  • ...and 7 more figures

Theorems & Definitions (13)

  • Theorem 1: Perron-Frobenius Theorem
  • Theorem 2: Collatz-Wielandt Formula
  • Definition 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 3 more