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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $a_1, \\dots, a_n$ be nonnegative integers.\nIn the [[ring of Laurent polynomials|1R5P656GS6]] $\\Q[x_1^{\\pm 1}, \\dots, x_n^{\\pm 1}]$, the constant term of\n$$\\prod_{1 \\le i \\ne j \\le n} \\left( 1 - \\frac{x_i}{x_j} \\right)^{a_i}$$\nis the multinomial coefficient\n$$\\frac{(a_1 + a_2 + \\cdots + a_n)!}{a_1!\\, a_2! \\cdots a_n!}.$$\n\n## Remarks\n\nFreeman Dyson stated this as a conjecture in 1962, in work on the statistical theory of energy levels.\nIt was proved the same year by Gunson and by Wilson, and Good gave a short proof in 1970.\nGood's proof shows that both sides satisfy the recurrence $F(a) = \\sum_{i} F(a - e_i)$ when all $a_i \\ge 1$, and that both reduce to the case of $n - 1$ variables when some $a_i = 0$.\nThe recurrence for the left side comes from the identity $\\sum_{i=1}^n \\prod_{j \\ne i} \\left(1 - \\frac{x_j}{x_i}\\right)^{-1} = 1$, a case of Lagrange interpolation.\nExample: $n = 2$.\nThe constant term of $(1 - x_1/x_2)^{a_1} (1 - x_2/x_1)^{a_2}$ collects the products of the terms $(x_1/x_2)^k$ and $(x_2/x_1)^k$, and equals $\\sum_{k} \\binom{a_1}{k}\\binom{a_2}{k} = \\binom{a_1 + a_2}{a_1}$ by [[Vandermonde's identity|01JADB9VW5]].\n\n## Explanations\n\n### L1 human\n\nMultiply out a product of many factors of the form $1 - x_i/x_j$ and keep only the terms with no variable left.\nThe Dyson conjecture, now a theorem, says that the sum of these terms is a multinomial coefficient, the number of ways to arrange $a_1 + \\cdots + a_n$ objects of $n$ colours.\n\n### L2 human\n\nThe case where all $a_i$ equal $k$ gives $\\frac{(nk)!}{(k!)^n}$, which computes the normalizing constant of the circular ensembles of [[random matrices|1HS4RT7HK5]] through an integral over the torus.\nAndrews proposed a $q$-analogue, proved by Zeilberger and Bressoud.\nThe equal-parameter case is the type $A$ case of the [[Macdonald constant term conjecture|5J25W265JA]].\n",
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