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  "body": "\n## Definition\n\nLet $P$ be a finite [[graded poset|1C7TKGTB2F]] with rank function $\\rho$.\nAssume that $P$ is [[bounded|B0QT3SM5DT]], with a least element $\\hat 0$ and a greatest element $\\hat 1$.\nFor $x \\le y$ in $P$, let $[x, y] = \\set{z \\in P : x \\le z \\le y}$.\nThe poset $P$ is *Eulerian* when for all $x \u003c y$ in $P$, the interval $[x, y]$ has as many elements of even rank as of odd rank.\n\n## Remarks\n\nEquivalently, the [[Möbius function|2Y75S8NG5B]] of $P$ is $\\mu(x, y) = (-1)^{\\rho(y) - \\rho(x)}$ for all $x \\le y$.\n\nExample: the lattice of subsets of a finite set $S$, with $\\rho(A) = \\abs{A}$.\nCheck: for $A \\subsetneq B$ with $\\abs{B \\setminus A} = k \\ge 1$, the interval $[A, B]$ has $\\sum_{j \\text{ even}} \\binom{k}{j} = 2^{k-1}$ elements of each parity of rank.\n\nNon-example: the chain $0 \u003c 1 \u003c 2$ with $\\rho(i) = i$ is graded and bounded, but the interval $[0, 2]$ has two elements of even rank and one of odd rank.\n\nEvery interval of an Eulerian poset is again Eulerian.\nThe face lattice of a convex polytope is Eulerian.\n",
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