Growing Avoiders from the Right: An Operator-Theoretic Approach

Abstract: 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 that stays in the ``grow from the right'' world of enumerative combinatorics. A $v$-avoiding permutation is built by right insertion, we keep a pruned family of forbidden ranks (the \emph{frontier}), and the insertion step becomes a nonnegative transfer operator on a doubly weighted $\ell^\infty$ space. A length--quadratic penalty makes this operator a finite-rank perturbation of a contraction, so a standard core/tail argument \cite{IonescuTulceaMarinescu1950,Hennion1993} yields quasi-compactness and analyticity of the growth series, hence finite exponential growth for $\Av(v)$. The same operator skeleton applies, with the same bounded-frontier check, to two natural extensions that are visible to right-to-left growth: (i) right-anchored vincular patterns \cite{BabsonSteingrimsson2000}, and (ii) mesh patterns \cite{BrandenClaesson2011} whose shaded cells lie in a fixed right strip. This appears to give the first general Stanley--Wilf type statements for these two subclasses. Our formulation is completely internal -- we never pass to $0$--$1$ matrices -- and it isolates the single pattern-dependent step (a bounded frontier increment) from the rest of the operator-theoretic machinery.