Transformations with Even Injective Restrictions
Overview
For a map on a finite ordered set, an inversion is a pair of inputs whose images appear in the opposite order. The paper classifies maps for which every choice of t inputs with distinct images has an even number of inversions. Inputs sharing an image form a block, and the number of blocks is the rank. The resulting block conditions answer Vítor H. Fernandes's two missing-rank questions, at ranks t and t+1, in arXiv 2605.12342. The paper gives closed counts for these maps and for maps unchanged by applying them twice, and extends the classification to partially defined maps and to restrictions of odd parity.
Original abstract (English)
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.
Original public questions
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.
Mathematical review
The manuscript has completed internal review. Complete formalization is not yet established for this public version.
Review standard