单射限制均为偶的变换
Transformations with Even Injective Restrictions
研究概述
对有限有序集上的映射，逆序是指一对输入的像以相反次序排列。本文分类如下映射：任取 t 个像互不相同的输入，逆序数都为偶数。具有相同像的输入组成一个块，块数称为秩。所得分块条件回答了 Vítor H. Fernandes 在 arXiv 2605.12342 中提出的两个缺失秩层问题，即秩为 t 和 t+1 的情形。论文给出这些映射及连续作用两次仍与作用一次相同的映射的封闭计数公式，并将分类扩展到部分定义的映射及逆序数为奇数的限制。
原文摘要（英文）
We classify the transformations of a finite chain for which every injective restriction to a prescribed number of points is even. The classification separates the kernel partition from the induced permutation of its image. At the critical rank, the kernels are precisely the noncrossing partitions whose gaps between consecutive points of a block contain an even number of whole blocks. Above that rank, the kernels are interval partitions or cyclic-interval partitions, according to the parity of the restriction size. This answers the two missing-rank questions posed by Fernandes. We derive closed formulas for the number of critical kernels, the number of maps in every rank layer, and the number of idempotents. An extension lemma also establishes regularity and describes the Green classes and maximal subgroups. The classification and its enumerative consequences follow from the parity change caused by replacing one representative of a kernel block.
原始公开问题
Vítor H. Fernandes: Groups of permutations that are even on maximal proper subsets, and related monoids — arXiv 2605.12342v1, page 2: the two unnumbered missing-rank questions for Sigma_n^t at ranks t and t+1, 2 ≤ t ≤ n-2.
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