Growing Avoiders from the Right: An Operator-Theoretic Approach

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.