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arXiv · 2608.11085

Bijections between pattern-avoiding derangements and desarrangements

Alyssa G. Henke·Derek H. Stephens·Yan Zhuang
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Abstract

Derangements are permutations without fixed points, and are in bijection with desarrangements: permutations whose first non-descent is even, or equivalently, permutations without ``pixed points''. Bsila, Cox, Hugo, Styron, and Zhuang recently proved a theorem characterizing all $\Pi\subseteq\mathfrak{S}_{3}$, such that $1\leq\left|\Pi\right|\leq3$, for which the number of derangements avoiding all patterns in $\Pi$ is equal to the number of desarrangements avoiding all patterns in $\Pi$. They left finding a bijective proof of this theorem as an open problem, and posed a related conjecture concerning the distributions of fixed points and pixed points over pattern avoidance classes. In this paper, we give bijective proofs of this theorem and conjecture.

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Alyssa G. Henke, Derek H. Stephens, Yan Zhuang. 2026-08-11. Bijections between pattern-avoiding derangements and desarrangements. https://arxiv.org/abs/2608.11085

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