AI 中文总结
本文针对Bsila等人提出的、关于避免模式的错排与非全降排列数量相等的定理及相关猜想，给出了其双射证明。
AI 中文摘要
错排是没有不动点的排列，与非全降排列构成双射：非全降排列指第一个非降点为偶数的排列，等价于没有“pixed点”的排列。Bsila、Cox、Hugo、Styron和Zhuang近期证明了一个定理，刻画所有满足1≤|Π|≤3的Π⊆𝔖₃，使得避免Π中所有模式的错排数量等于避免Π中所有模式的非全降排列数量。他们将该定理的双射证明留作开放问题，并提出了关于不动点与pixed点在模式避免类上分布的相关猜想。本文给出该定理及猜想的双射证明。
英文摘要
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 $Π\subseteq\mathfrak{S}_{3}$, such that $1\leq\left|Π\right|\leq3$, for which the number of derangements avoiding all patterns in $Π$ is equal to the number of desarrangements avoiding all patterns in $Π$. 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.
Comments21 pages
