English

Bijections between pattern-avoiding derangements and desarrangements

Combinatorics 2026-08-11 v1

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 ΠS3\Pi\subseteq\mathfrak{S}_{3}, such that 1Π31\leq\left|\Pi\right|\leq3, for which the number of derangements avoiding all patterns in Π\Pi is equal to the number of desarrangements avoiding all patterns in Π\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.

Keywords

Cite

@article{arxiv.2608.11085,
  title  = {Bijections between pattern-avoiding derangements and desarrangements},
  author = {Alyssa G. Henke and Derek H. Stephens and Yan Zhuang},
  journal= {arXiv preprint arXiv:2608.11085},
  year   = {2026}
}

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21 pages