A desarrangement is a permutation whose first ascent is even. The first author introduced this class of permutations for the purpose of combinatorially explaining a standard recurrence relation for the number of derangements of n elements. He gave a nice bijection between derangements and desarrangements. Here we extend his bijection to obtain more refined enumeration results relating derangements and desarrangements. In particular, we construct a bijection between descent classes of derangements and descent classes of desarrangements. The following monotonicity result is a consequence of this correspondence: In any (nontrivial) descent class the number of permutations with exactly k fixed points decreases as k increases.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics