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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chloe!
@schala163
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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chloe!
@schala163
but the issue is like... only one of the terms contributes to the growth rate. And while it's easy to categorify something like x^2+x-1, and maybe plausible to categorify "some expression involving the roots of that thing which magically has all the radicals cancel"...
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chloe!
@schala163
i don't see how to get past that point.. like, how can we tell the difference between the bigger root and the littler one? which is the only thing that actually matters??? IDK :shrug:
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