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  "body": "\n## Summary\n\nAnalytic combinatorics finds the asymptotic size of combinatorial counts from their [[generating functions|0PHAC833JF]], see the [[definition|XGZ2YYCNWK]].\nIt views a generating function as a function of a complex variable.\nThe singularities closest to the origin decide the exponential growth of the coefficients, and the type of these singularities decides the correction factor.\nTogether with the [[symbolic method|JP11D90EJC]], it is the subject of the book \"Analytic Combinatorics\" by Philippe Flajolet and Robert Sedgewick.\n\n## Explanations\n\n### L1 human\n\nA generating function packs a whole sequence of counts into one function.\nIf one plugs complex numbers into that function, it blows up or misbehaves at some point.\nThe position of the nearest bad point tells how fast the counts grow.\nFor example, the Catalan generating function first breaks down at $z = 1/4$, so the Catalan numbers grow roughly like $4^n$.\nThe shape of the breakdown, here a square root, gives the extra factor $n^{-3/2}$.\n",
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