http://bit-player.org/2006/best-friends An amateur's outlook on computation and mathematics. Fri, 29 Aug 2008 05:13:13 +0000 http://wordpress.org/?v=1.5.2 http://bit-player.org/2006/best-friends#comment-9 Mon, 30 Jan 2006 22:26:09 +0000 http://bit-player.org/2006/best-friends#comment-9 The one-dimensional version should translate into a problem about random permutations. Once <em>n</em> + 1 numbers <em>x</em>₀ ,..., <em>x_n</em> have been chosen, say in [0,1], all that really matters is the ranking of the <em>n</em> distances <em>x</em>₁ - <em>x</em>₀, <em>x</em>₂ - <em>x</em>₁ ,..., <em>x_n</em> - <em>x</em>_{<em>n</em> - 1}, which will be a permutation of 1, 2 ,..., <em>n</em>. If 1 corresponds to the shortest distance and <em>n</em> to the largest, then each best-friend pair corresponds to a number in the permutation that is smaller than the numbers on either side of it. The one-dimensional version should translate into a problem about random permutations. Once n + 1 numbers x₀ ,…, x_n have been chosen, say in [0,1], all that really matters is the ranking of the n distances x₁ - x₀, x₂ - x₁ ,…, x_n - x_{n - 1}, which will be a permutation of 1, 2 ,…, n. If 1 corresponds to the shortest distance and n to the largest, then each best-friend pair corresponds to a number in the permutation that is smaller than the numbers on either side of it.

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