Conversation

chloe!
@schala163
if i have some sequence a_n=number of ways to do some nonsense to {1,...,n} then i might wanna look at the formal power series sum a_nx^n or maybe stick a 1/n! in there... and lots of combinatorial operations i might care about correspond to natural operations on these... 1/?
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chloe!
@schala163
and there's this theory of combinatorial species that gives a really wonderful categorification of this whole yoga. It's really great, you can view this all a shadow of some much more combinatorially meaningful operations by just being a little bit category brained... 2/???
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chloe!
@schala163
but okay a lot of the time the actual *reason* i wanna work with formal power series is... i wanna *estimate* a_n. And it turns out that you can read off lots of stuff about the growth of a_n from the analytic properties of the *function represented by the power series*... 3/idk
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chloe!
@schala163
this is dope af of course, but... what the heck does it *mean*??? does the species story have anything to say about the fact that the growth rate of a_n is controlled by the poles of the analytic function i get by *evaluating* its power series at *complex numbers* are? 4/shrug
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chloe!
@schala163
ive wondered about this a lot and afaict the answer is "not a chance" and yet... idk there's GOTTA BE SOMETHING COME ON 5/5 im literally crying
@QiaochuYuan
this is your shit right save me
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QC (in Berlin 7/29-??? w/ COVID 😷)
@QiaochuYuan
this is a good question and i've sort of wondered fruitlessly about similar questions. a special case that i think is already pretty hopeless is: is there in any reasonable sense a "bijective" or "categorical" way to understand the asymptotics of the fibonacci numbers?
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QC (in Berlin 7/29-??? w/ COVID 😷)
@QiaochuYuan
one of the only examples i even kinda have a handle on is stirling's formula: you can give a conceptual proof that e^n \ge n^n/n! using groupoid cardinality (LHS is groupoid of sets equipped with a map to [n], RHS is sets of cardinality n equipped with a map to [n])
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QC (in Berlin 7/29-??? w/ COVID 😷)
@QiaochuYuan
but it doesn't buy you all that much over just the really obvious saddle point bound, which is just saying that e^n is greater than or equal to a term in its taylor series... and i think getting the sqrt{2 pi n} factor from arguments like this is probably hopeless
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