Counting nilpotent endomorphisms

Abstract

A variant of Prüfer's classical proof of Cayley's 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 F q is q n ( n - 1 ) .