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  "body": "\n## Definition\n\nLet $\\Delta$ be a finite [[simplicial complex|V8GZ0W1ZZ5]] of dimension $d - 1$, with $d \\ge 1$.\nFor $-1 \\le i \\le d - 1$, let $f_i$ be the number of faces of $\\Delta$ of dimension $i$, that is, with $i + 1$ vertices; so $f_{-1} = 1$ counts the empty face.\nThe *f-vector* of $\\Delta$ is $(f_{-1}, f_0, \\dots, f_{d-1})$.\nThe *h-vector* of $\\Delta$ is the sequence $(h_0, h_1, \\dots, h_d)$ defined by\n$$\\sum_{i=0}^{d} h_i \\, t^{d-i} = \\sum_{i=0}^{d} f_{i-1} \\, (t - 1)^{d-i}.$$\n\n## Remarks\n\nComparing coefficients gives $h_k = \\sum_{i=0}^{k} (-1)^{k-i} \\binom{d-i}{k-i} f_{i-1}$.\nIn particular $h_0 = 1$, $h_1 = f_0 - d$, and $h_0 + h_1 + \\dots + h_d = f_{d-1}$.\nThe f-vector can be recovered from the h-vector, so both carry the same information.\nFor a simplicial polytope of dimension $d$ one uses the f-vector and h-vector of its boundary complex, which has dimension $d - 1$.\n\nExample: the boundary of an octahedron has $d = 3$ and $f = (1, 6, 12, 8)$.\nThen $(t-1)^3 + 6(t-1)^2 + 12(t-1) + 8 = t^3 + 3t^2 + 3t + 1$, so $h = (1, 3, 3, 1)$, and indeed $1 + 3 + 3 + 1 = 8 = f_2$.\n\nNon-example of a nice property: the h-vector can have negative entries.\nFor two disjoint edges, $d = 2$ and $f = (1, 4, 2)$, and $(t-1)^2 + 4(t-1) + 2 = t^2 + 2t - 1$ gives $h = (1, 2, -1)$.\n",
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