The largest and the smallest fixed points of permutations

We give a new interpretation of the derangement numbers dndn as the sum of the values of the largest fixed points of all non-derangements of length n−1n−1. We also show that the analogous sum for the smallest fixed points equals the number of permutations of length nn with at least two fixed points. We provide analytic and bijective proofs of both results, as well as a new recurrence for the derangement numbers.