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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
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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chloe!
@schala163
Replying to
@QiaochuYuan
hmm yeah okay so there is an explicit formula for the n'th fibonacci number. usually u get it by mucking around with the generating function, do a partial fraction expansion blah blah everything up to this point seems like it could plausibly categorify nicely...
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chloe!
@schala163
iirc there's nice combinatorially explicit way to see how when u expand out the binet formula w/ a binomial expansion on each of the (1+/-sqrt{5})^n terms it spits out the fibonacci numbers. something about tilings i forget how it works but i swear there's a nice way...
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