The Probability That an Operator Is Nilpotent

Choose a random linear operator on vector space of finite cardinality N; then the probability that it is nilpotent 1/N. This analogue fact for self-map set N, some iterate constant The first result due to Fine, Herstein, and Hall, second essentially Cayley’s tree formula. We give new proof nilpotents, analogous Joyal’s beautiful It uses only general algebra avoids calculation entirely.