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  "body": "\n## Statement\n\nLet *X* be a finite CW complex, of dimension *d*, with Poincaré polynomial\n\n$$P_X (t) = \\sum_{k=0}^d b_k(X) \\, t^k$$\n\nwhere $b_k(X)$ is its *k*th Betti number. Then the Poincaré polynomial of the *n*th symmetric product of *X*, denoted $\\operatorname{Sym}^n (X)$, is obtained from the series expansion:\n\n$$\\operatorname{PE}[P_X(-t)\\,x] = \\prod_{k=0}^d (1-t^k x)^{(-1)^{k+1}b_k(X)} = \\sum_{n\\geq 0} P_{\\operatorname{Sym}^n(X)}(-t) \\, x^n$$\n",
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        "locator": "Plethystic exponential, section Macdonald's formula for symmetric products",
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  "name": "Macdonald's formula for symmetric products",
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