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  "body": "\n## Definition\n\nLet $N$ and $X$ be finite sets with $\\abs{N} = n$ and $\\abs{X} = x$.\nConsider the [[functions|0XHDTE5YMZ]] $f \\colon N \\to X$ in one of three classes: all functions, [[injective functions|0PVG1CRAJN]], or [[surjective functions|0XKH881BZ7]].\nCount them in one of four ways:\n\n1. as functions;\n2. up to permutations of $N$, so $f$ and $f \\circ \\sigma$ count as the same for every permutation $\\sigma$ of $N$;\n3. up to permutations of $X$, so $f$ and $\\tau \\circ f$ count as the same for every permutation $\\tau$ of $X$;\n4. up to permutations of both, so $f$ and $\\tau \\circ f \\circ \\sigma$ count as the same for all such $\\sigma$ and $\\tau$.\n\nThe *twelvefold way* is the list of these $3 \\cdot 4 = 12$ counting problems with their answers.\n\n## Remarks\n\nWrite $x^{\\underline{n}}$ for the [[falling factorial|0N5M924N3P]], $S(n, k)$ for the [[Stirling numbers of the second kind|0JCQGPY8Z4]], and $p_k(m)$ for the number of [[partitions|0JV30XN73Z]] of $m$ into exactly $k$ parts.\nWrite $[n \\le x]$ for $1$ if $n \\le x$ and $0$ otherwise.\nThe answers, with $n$ balls $N$ and $x$ boxes $X$, are as follows.\n\n| count | all functions | injective | surjective |\n|---|---|---|---|\n| as functions | $x^n$ | $x^{\\underline{n}}$ | $x!\\, S(n, x)$ |\n| up to permutations of $N$ | $\\binom{x + n - 1}{n}$ | $\\binom{x}{n}$ | $\\binom{n - 1}{x - 1}$ |\n| up to permutations of $X$ | $\\sum_{k=0}^{x} S(n, k)$ | $[n \\le x]$ | $S(n, x)$ |\n| up to both | $\\sum_{k=0}^{x} p_k(n)$ | $[n \\le x]$ | $p_x(n)$ |\n\nThe entry $\\binom{n-1}{x-1}$ needs $n \\ge 1$; for $n = x = 0$ there is exactly one surjection, the empty map.\n\nExample: take $n = 3$ and $x = 2$, counted up to permutations of $N$.\nA class of functions is then fixed by how many balls go into each box, so the classes are $(3, 0)$, $(2, 1)$, $(1, 2)$ and $(0, 3)$.\nThis gives $4 = \\binom{4}{3}$, as in the table.\n\nNon-example: counting functions up to a permutation of $N$ that is not arbitrary, for instance only up to cyclic rotations of $N$, is not one of the twelve problems.\nIt leads to necklace counting, which is handled by the [[Pólya enumeration theorem|MPYQ39P01F]].\n\nThe second row is the theory of [[combinations with repetition|0VQQCVG8WA]].\nThe first entry of the first row is the [[number of functions between finite sets|08QQH9F7G0]], and the second is the [[number of injections|0X0N833FF8]].\n",
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