## December 30, 2019

### 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:

Posted at December 30, 2019 7:10 PM UTC

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### 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).

Posted by: Todd Trimble on December 31, 2019 1:29 PM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

I’m sorry; I really should have checked first. Other results in the paper are attributed to Labelle (and others), but the Cayley formula proof is apparently Joyal’s.

There was some interesting discussion about Rota and generating functions at MathOverflow, including this mention by Tony Huynh about species. The rest of my comment above was influenced by this.

Posted by: Todd Trimble on December 31, 2019 2:08 PM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

I admit, when I wrote “Joyal’s wonderful proof of Cayley’s formula that led him (I guess) to create the equally wonderful theory of species”, I was completely speculating. It’s easy to imagine how that could have been the case, but that doesn’t mean it’s true!

Maybe I should also have mentioned that Joyal’s proof is one of the proofs from the Proofs from the Book book.

Posted by: Tom Leinster on January 1, 2020 2:39 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

Nice, Tom! I hope you try to publish this. I think people prefer short papers, and I think referees are a lot like people in this respect.

It might be nice to pound in your point by including a Corollary to Theorem 3.1 where you conclude that #Nil(X)/#Lin(X) = 1/#X. Right now it’s sort of like you’re playing the whole symphony except the crashing final chord (because it’s “too obvious”). The only place you mention this fact is in the first sentence of the abstract and the first paragraph of the paper.

Posted by: John Baez on December 31, 2019 6:59 PM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

Thanks, John. And you may very well be right. But in an uncharacteristic turn of efficiency, I already submitted it!

I’ll take this opportunity to get something off my chest. When I was choosing a journal, I came across one that seemed to be perfect: right subject area, someone on the editorial board who’s written on this subject before, diamond open access after a battle with a big predatory publisher… 10/10. But then I saw a killer phrase in the journal’s self-description: “the journal does not typically publish new proofs of known results”. That is of course absolutely their choice to make, but it makes me realize (again) that there are some mathematicians whose priorities I simply do not understand. I read it as “we only really care about which things are true and which things are false. We typically don’t care why things are true or false.” And I have a hard time understanding how someone can be a mathematician and feel that way.

Posted by: Tom Leinster on January 1, 2020 2:47 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

They could be under the delusion that once you have proved something, you’re done figuring out why it’s true. A proof is indeed a reason why something is true. But the first proof is often just a preliminary scouting expedition.

Posted by: John Baez on January 1, 2020 7:13 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

This reminds me of a bit you quoted from Peter Cameron’s blog:

There are very many different proofs of this theorem [Cayley’s]; each proof tells us something new. It is one of the strongest arguments I know for having many proofs of a theorem.

Funnily enough, Peter is an editor of the journal I’m talking about! I guess there’s no point being coy: it’s Algebraic Combinatorics.

Posted by: Tom Leinster on January 1, 2020 11:53 PM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

Terry Tao has a habit of saying every journal he submits papers to, when writing about his preprints on his blog.

Posted by: David Roberts on January 4, 2020 12:53 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

Interesting; I wonder why. Of course, Terry Tao is a very special case. I suspect rejection for him is a much less common occurrence than for most of us.

All I’ve done here is disclose the name of a journal I didn’t submit to — although I would have liked to.

Posted by: Tom Leinster on January 4, 2020 1:02 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

I’m reminded of a line from Eugenia Cheng’s How to Bake $\pi$:

When I read someone else’s math, I always hope the author will have included a reason and not just a proof.

Posted by: Blake Stacey on January 2, 2020 2:31 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

Perhaps, in fairness to the journal and its editors, it is important to note that “we don’t publish new proofs of known results” is not the same as “we don’t appreciate new proofs of known results.” After all, there are plenty of journals that don’t publish expository papers, and I think few believe that means that their editors don’t believe in the values of expository papers.

Posted by: L Spice on January 2, 2020 2:37 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

Sort of like my wife’s dad used to say: “I like spinach — just not enough to eat it”.

Posted by: John Baez on January 2, 2020 7:48 AM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

The word typically in their instructions is frustratingly vague. Whenever I read a journal’s “About Us” or “Aims and Scope”, I wonder how much of it represents a well-thought-out consensus of the leadership and how much was just thrown in by one person and then overlooked because it resembled every other “About Us” or “Aims and Scope” webpage.

The line between new and old can get contentious, and edicts set out in policy can turn out more fluid in practice. There’s a moderately famous bit of lore that Physical Review Letters rejected the first paper about lasers because they thought it was just another maser paper. Like much physics lore, this might even be halfway true. The editor of PRL, Samuel Goudsmit, actually declared a subject non grata three times in its first decade: masers in 1959, the Mössbauer effect in 1960 and gauge theories in 1965. In all these cases, the motivation was that too many mediocre, trend-following papers were being received, and so new ones would only be considered if they stuck to a high standard of concreteness (“if they propose a genuinely new concept which makes possible the correlation of previously unrelated data”). Later, in 1973, Goudsmit wrote that the Physical Review journals were receiving “papers about fundamental theories” for which it was impossible to find willing reviewers. These submissions, he noted, showed “a paucity of mathematics as compared to wordage” and refashioned old theories without predicting new experimental consequences. He then gave the genre a polite brush-off:

Some of these papers may have an important bearing on the philosophy of physics. However, since there exist excellent journals publishing articles on the foundations and on the philosophy of science, we shall no longer accept papers of this type for the Physical Review.

Still, the prohibition on foundational matters was not absolute, and for example, Physical Review D would accept a paper by d’Espagnat on Bell inequalities the following year.

I wonder how much prohibitions on new proofs of old theorems contribute to the phenomenon that Davis and Hersh noted in their essay “The Ideal Mathematician”. The writings of such an individual

follow an unbreakable convention: to conceal any sign that the author or the intended reader is a human being.

Posted by: Blake Stacey on January 3, 2020 8:02 PM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

L Spice wrote:

Perhaps, in fairness to the journal and its editors, it is important to note that “we don’t publish new proofs of known results” is not the same as “we don’t appreciate new proofs of known results.”

I agree, Loren, that they’re not the same. Plus, as I said, the journal’s declaration is that it “does not typically publish new proofs of known results” (my emphasis) — it’s not a hard and fast rule. And moreover, as Blake mentioned, the policies of an institution don’t always faithfully reflect the preferences of the human beings behind it (for all sorts of reasons, but principally that everyone’s busy).

Nevertheless, John’s spinach anecdote is spot on. Claims to appreciate X or find Y important are unconvincing if you don’t actually take action towards X or Y. (If I claim climate change is important to me, but e.g. vote for a political party with regressive environmental policies, then you can question how genuine my claim really is.) So actually, when you write this —

there are plenty of journals that don’t publish expository papers, and I think few believe that means that their editors don’t believe in the values of expository papers

— I kind of know what you mean (institutional inertia etc.), but at the same time I might count myself as one of your “few”. Any journal editor who really believes in the value of expository papers — anyone who’s reached the conclusions that mathematics needs more expository writing, that expository writing is undervalued in terms of prestige and careers, and that there’s a shortage of good venues for publishing expository pieces — would make the small effort required to start a discussion of their journal’s policy. Otherwise it’s only “belief” in the weak sense of lip service, not “belief” that’s strong enough to actually cause action. It’s like John’s wife’s dad’s attitude to spinach.

Posted by: Tom Leinster on January 4, 2020 8:39 PM | Permalink | Reply to this

### Re: Counting Nilpotents: A Short Paper

By the way, Tom, I think all the stuff I’ve been doing on random permutations — especially the stuff involving species and groupoid cardinality — deserves to be generalized to finite vector spaces. I guess the analogue of the cycle decomposition of a permutation is the rational canonical form. Since this involve more parameters than just a natural number, everything may become a bit more fancy (unless we just ignore those parameters).

Posted by: John Baez on December 31, 2019 7:11 PM | Permalink | Reply to this

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