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arXiv · 2607.21466

A lower bound on the growth rate of $(132,213)$-avoiding cyclic permutations

Robert Laudone
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Abstract

We construct a new reduction process which takes a $(132,213)$-avoiding permutation to a shorter one that is cyclic if and only if the original was. Iterating it determines whether a given $(132,213)$-avoiding permutation is cyclic. Reversing it gives four moves that build every cyclic $(132,213)$-avoiding permutation, uniquely, from $1$ if $n$ is odd, and $21$ if $n$ is even. Our main application is the first non-trivial lower bound for the growth rate of $\mathcal{C}_n(132,213)$, the cyclic permutations of length $n$ avoiding $132$ and $213$. We also give several other consequences of the reduction, including a bijection between the odd and even size classes and an exact enumeration for those permutations with a restricted number of layers.

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Robert Laudone. 2026-07-23. A lower bound on the growth rate of $(132,213)$-avoiding cyclic permutations. https://arxiv.org/abs/2607.21466

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