Growing Avoiders from the Right: An Operator-Theoretic Approach

TL;DR

This paper presents an operator-theoretic proof of the Stanley--Wilf conjecture for permutation patterns, using a novel internal approach that avoids $0$--$1$ matrices and leverages transfer operators to analyze growth.

Contribution

It introduces a new internal, operator-theoretic framework for permutation pattern avoidance, providing a parallel proof of the Stanley--Wilf conjecture without relying on matrix reduction.

Findings

Established bounded transfer operators imply finite exponential growth.

Provided an abstract theorem linking frontier growth and transfer operator bounds.

Achieved analyticity of the growth series through a Neumann-series argument.

Abstract

(Work in progress) Marcus and Tardos \cite{MarcusTardos2004} proved the Stanley--Wilf conjecture by reducing pattern avoidance to an extremal problem on $0$--$1$ matrices. We give a parallel proof for classical permutation patterns that stays entirely in the ``grow from the right'' world of enumerative combinatorics. A $v$-avoiding permutation is built by right insertion; at each step we keep a pruned family of locations of $(k{-}1)$-partial occurrences of $v$ (the \emph{frontier}), each carrying its forbidden rank interval. The insertion step then induces a nonnegative transfer operator on a doubly weighted $\ell^\infty$ space. A quadratic penalty in the length makes this operator bounded, and a Neumann-series argument on a natural separable predual yields analyticity of the growth series, hence finite exponential growth for $\Av(v)$. The formulation is completely internal -- we never pass to $0$--$1$ matrices -- and it cleanly separates the pattern-dependent combinatorics of the frontier from a purely operator-theoretic core. In particular, we obtain an abstract ``right-insertion/transfer-operator'' theorem: any system whose frontier grows at most linearly and whose transfer operator satisfies a uniform quadratic length bound has an analytic growth series.