In the past months, I spent as much time as I had on taking online courses at Coursera. One particularly interesting course, both from a mathematical and computational point of view, is Analytic CombinatoricsÂ which applies combinatorics (i.e., the art of counting) to the analysis of algorithms by finding formulae, exact or asymptotic, for their running time.

It is notoriously difficult to find exactÂ formulae for general combinatorial constructs. Typically, we want to know how many objects, e.g., trees, permutations, sequences, with certain properties there are of a given size. Famously, the number of binary trees (and about a million other constructions) is governed by the Catalan numbers

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## Everything You Always Wanted to Know About Primes (But Were Afraid to Ask)