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  "body": "\n## Definition\n\nLet $(S_n)_{n \\ge 0}$ be a family of [[finite sets|0BBTH3YSZA]] given by a combinatorial rule, such as the subsets of $\\set{1, \\ldots, n}$ or the trees with $n$ vertices.\n*Enumerative combinatorics* is the branch of [[combinatorics|0JKW7W7XFR]] that determines the counting function $n \\mapsto \\abs{S_n}$.\nAn answer is one of the following:\n\n1. an explicit formula for $\\abs{S_n}$;\n2. a [[recurrence relation|0FYX44GP92]] that computes $\\abs{S_n}$ from earlier values;\n3. a closed form for a [[generating function|0PHAC833JF]] of the sequence $\\abs{S_n}$;\n4. an asymptotic estimate of $\\abs{S_n}$ as $n \\to \\infty$.\n\n## Remarks\n\nExample: let $S_n$ be the set of subsets of $\\set{1, \\ldots, n}$ with no two consecutive elements.\nA set in $S_n$ either avoids $n$, or contains $n$ and avoids $n - 1$.\nThis gives the recurrence $\\abs{S_n} = \\abs{S_{n-1}} + \\abs{S_{n-2}}$, with $\\abs{S_0} = 1$ and $\\abs{S_1} = 2$, so $\\abs{S_n}$ is the Fibonacci number $F_{n+2}$.\n\nNon-example: the question whether a finite projective plane of order $10$ exists is a question of existence, not of counting.\nIt belongs to the theory of [[combinatorial designs|PAJTMQ604P]].\n\nThe parameter $n$ may be replaced by several parameters, as in the [[binomial coefficient|0GBZM0MT0F]] $\\binom{n}{k}$.\n",
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