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 , such that , for which the number of derangements avoiding all patterns in is equal to the number of desarrangements avoiding all patterns in . 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.
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