Abstract:
Stanley conjectured that the number of alternating permutations on $[2n]$ with maximal number of fixed points equals the number of derangements on $[n]$. While this is not too hard to prove, it can also be extended to the following: Permutations on $[n]$ with $k$ falling blocks (containing only descents) of length $a_1, a_2, \dots, a_k$ and maximal number of fixed points are in bijection with derangements on $[n-k]$ with falling blocks of length $a_1-1, a_2-1, \dots, a_k-1$, and possibly with descents between blocks.