The largest and the smallest fixed points of permutations

We give a new interpretation of the derangement numbers dn as the sum of the values of the largest fixed points of all non-derangements of length n− 1. We also show that the analogous sum for the smallest fixed points equals the number of permutations of length n 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. 1 Largest fixed point Let [n] = {1, 2, . . . , n}, and let Sn denote the set of permutations of [n]. Throughout the paper, we will represent permutations using cycle notation unless specifically stated otherwise. Recall that i is a fixed point of π ∈ Sn if π(i) = i. Denote by Dn the set of derangements of [n], i.e., permutations with no fixed points, and let dn = |Dn|. Given π ∈ Sn \ Dn, let l(π) denote the largest fixed point of π. Let an,k = |{π ∈ Sn : l(π) = k}|. Clearly, an,1 = dn−1 and an,n = (n− 1)!. (1) It also follows from the definition that