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  "body": "\n## Statement\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] in which every set $\\set{y \\in P : y \\le x}$, for $x \\in P$, is finite.\nLet $\\mu$ be the Möbius function of $P$: $\\mu(x, x) = 1$, $\\mu(x, y) = -\\sum_{x \\le z \u003c y} \\mu(x, z)$ for $x \u003c y$, and $\\mu(x, y) = 0$ when $x \\not\\le y$.\nLet $K$ be a [[commutative ring|0310REJJY8]] and let $f, g \\colon P \\to K$.\nThen\n$$g(x) = \\sum_{y \\le x} f(y) \\quad \\text{for all } x \\in P$$\nif and only if\n$$f(x) = \\sum_{y \\le x} \\mu(y, x) \\, g(y) \\quad \\text{for all } x \\in P.$$\n\n## Remarks\n\nIn the [[incidence algebra|1H9SFWHYEB]] the Möbius function is the inverse of the zeta function $\\zeta$, which is $1$ on all pairs $y \\le x$.\nThe first equation says $g = f \\zeta$ and the second says $f = g \\mu$, where a function on $P$ is multiplied by an element of the incidence algebra on the right.\nSince $\\zeta \\mu = \\mu \\zeta = \\delta$, each equation implies the other.\nThere is a dual form with sums over $y \\ge x$, under the hypothesis that every set $\\set{y : y \\ge x}$ is finite.\n\nExample: for the subsets of a finite set ordered by inclusion, $\\mu(S, T) = (-1)^{\\abs{T \\setminus S}}$, and the theorem becomes the inclusion–exclusion principle.\nFor the positive integers ordered by divisibility, $\\mu(d, n)$ is the number-theoretic Möbius function of $n/d$, and the theorem becomes the classical [[Möbius inversion formula|07Q4TBTFVB]].\nFor the chain $0 \u003c 1 \u003c 2 \u003c \\dots$, $\\mu(x, x+1) = -1$ and $\\mu(x, y) = 0$ for $y \\ge x + 2$, so $f(x) = g(x) - g(x-1)$: a difference undoes a partial sum.\n\nThe finiteness hypothesis is needed for the sums to make sense: on $\\Z$ with its usual order, $\\sum_{y \\le x} f(y)$ is an infinite sum.\n\n## Explanations\n\n### L1 human\n\nIf you know the totals $g(x)$ obtained by adding $f$ over everything below $x$, you can recover $f$.\nThe recipe uses signed weights $\\mu(y, x)$ that depend only on the shape of the poset.\nFor ordinary numbers in a line it just says: take the difference of two consecutive running totals.\n\n### L2 human\n\nThe theorem was put in this general form by Rota, and it unifies many counting techniques.\nTo use it, compute $\\mu$ for the poset at hand, often with product formulas or with the recursion.\nFor the lattice of partitions of a set, $\\mu$ of the whole interval is $(-1)^{n-1}(n-1)!$, and inversion counts connected structures from all structures.\nIn the lattice of subspaces of $\\F_q^n$, $\\mu$ involves powers of $q$, and inversion counts subspaces avoiding given conditions.\n",
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