### Counting Nilpotents: A Short Paper

#### Posted by Tom Leinster

Inspired by John’s recent series of posts on random permutations, I started thinking about random operators on vector spaces, and nilpotent operators, and Cayley’s tree formula, and, especially, Joyal’s wonderful proof of Cayley’s formula that led him (I guess) to create the equally wonderful theory of species.

Blog posts and comments are often rambling and discursive. That’s part of the fun of it: we think out loud, we try out ideas, we stumble ignorantly through things that others have done better before us, we make mistakes, we refine our ideas, and we learn how to communicate those ideas more efficiently. My own posts on this topic (1, 2, 3) are no exception.

But short sharp accounts are also good! So I wrote a 4.5-page paper containing the thing I think is new. It’s a new proof of the old theorem that when you choose at random a linear operator on a vector space of finite cardinality $N$, the probability of it being nilpotent is $1/N$. And this proof is a linear analogue of Joyal’s proof of Cayley’s formula.

I’ve already written everything I want to on this topic, so I won’t say more, except to observe that for me, writing such a short paper feels unusual. I haven’t done it very often. I think I’ve written more 400-page books than 4-and-a-bit-page papers… more on which soon!

In any case, at least this time I’ve avoided the classic cycle:

## Re: Counting Nilpotents: A Short Paper

My memory could be tricking me, but I thought Joyal attributed the beautiful proof of Cayley’s formula to Gilbert Labelle. It could be that this was the key example that catalyzed his theory of species, but I can also easily imagine that the development came from multiple sources (perhaps it is significant that the founding papers were published in Rota’s journal, Advances in Mathematics).