Growing Avoiders from the Right: An Operator-Theoretic Approach
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918247459192832 |
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| author | Rastegar, Reza |
| author_facet | Rastegar, Reza |
| contents | (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growing Avoiders from the Right: An Operator-Theoretic Approach Rastegar, Reza Combinatorics (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. |
| title | Growing Avoiders from the Right: An Operator-Theoretic Approach |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.06118 |