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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $A$ be an $n \\times n$ [[doubly stochastic matrix|JJA90EB2PS]].\nThis means that its entries are nonnegative real numbers and that each row and each column sums to $1$.\nThen\n$$\\operatorname{perm}(A) \\ge \\frac{n!}{n^n},$$\nwhere $\\operatorname{perm}$ is the [[permanent|X9B1XETBPF]].\nEquality holds if and only if every entry of $A$ equals $1/n$.\n\n## Remarks\n\nThe statement was conjectured by B. L. van der Waerden and proved independently by G. P. Egorychev and D. I. Falikman.\nEgorychev's proof uses the Alexandrov–Fenchel inequality for mixed discriminants.\nA later short proof by L. Gurvits uses real stable polynomials.\n\nExample: for $n = 2$, every doubly stochastic matrix has the form $\\begin{pmatrix} t \u0026 1 - t \\\\ 1 - t \u0026 t \\end{pmatrix}$ with $0 \\le t \\le 1$.\nIts permanent is $t^2 + (1-t)^2$, whose minimum on $[0, 1]$ is $1/2 = 2!/2^2$, reached only at $t = 1/2$.\n\nThe value $n!/n^n$ is the permanent of the matrix with all entries $1/n$.\nBy Stirling's formula it is about $e^{-n} \\sqrt{2 \\pi n}$.\n\n## Explanations\n\n### L1 human\n\nA doubly stochastic matrix spreads one unit of mass in each row and each column.\nThe theorem says that its permanent cannot be very small, and that the most evenly spread matrix gives the smallest value.\n\n### L2 human\n\nFor a $k$-regular bipartite graph with $n$ vertices on each side, the biadjacency matrix divided by $k$ is doubly stochastic.\nSo the theorem gives at least $k^n n!/n^n$ perfect matchings.\nIn particular, every doubly stochastic matrix has a positive permanent, so it has a permutation pattern of positive entries.\n",
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  "msc": [
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  "name": "van der Waerden conjecture on permanents",
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