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The problem of estimating $B_n$ was posed by Knuth in 1992. Knuth conjectured that $b_n \\leq {n \\choose 2} + o(n^2)$ and also derived the first upper and lower bounds: $b_n \\leq 0.7924 (n^2 +n)$ and $b_n \\geq n^2/6 -O(n)$. 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