Naturally emerging maps for derangements and nonderangements

A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of $n$ elements, and we describe a bijective proof of this recurrence which can be found using a recursive map. We then show the combinatorial interpretation of this bijection and how it compares with other known bijections, and show how this gives an involution on $\mathfrak{S}_n$. Nonderangements satisfy a similar recurrence. We convert the bijective proof of the one-term identity for derangements into a bijective proof of the one-term identity for nonderangements.