johncarlosbaez, 14 days ago

To learn about combinatorial species, try any of the 3 free books linked to here:

https://math.ucr.edu/home/baez/permutations/

Or if you're really brave, just dive in and see how I apply them to random permutations. In learning about these, I learned something really surprising to me: random permutations are largely governed by the Poisson distribution. That got me interested in combinatorial proofs of Dobiński's formula. Besides the proof I just gave you, there's a nice proof due to Rota that uses Stirling numbers:

https://math.ucr.edu/home/baez/permutations/permutations_8.html

There's a lot of category theory in this business, so it's a fun mix of ideas.

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