Counting nilpotent endomorphisms

A variant of Prüferʹs classical proof of Cayleyʹs theorem on the enumeration of labelled trees counts the nilpotent self-maps of a pointed finite set. Essentially, the same argument can be used to establish the result of Fine and Herstein [Illinois J. Math. 2 (1958) 499–504] that the number of nilpotent n×n matrices over the finite field is qn(n-1).