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In how many ways can the numbers be arranged in an array ?

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October 5, 2010
linear algebra

Problem Statement

In how many ways can 1,2,…,2n1, 2,\ldots, 2n be arranged in a 2×n2 \times n rectangular array (a1a2⋯anb1b2⋯bn)\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right) for which: (i) a1<a2<⋯<an,a_1 < a_2 < \cdots < a_n,
(ii) b1<b2<⋯<bn,b_1 < b_2 <\cdots < b_n,
(iii) a1<b1,a2<b2,…,an<bn ?a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?
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