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Unformatted text preview: MAT 280 UC Davis, Winter 2011 Longest increasing subsequences and combinatorial probability Homework Set 2 Homework due: Wednesday 2/2/23 1. If f is a continual Young diagram and g is its rotated version, let K ( g ) = I hook ( f ) = Z Z f ( x ) log h f ( x,y ) dy dx be the hook integral that appears in the asymptotic version of the hook length formula d 2 | | ! = exp- n 1 + 2 I hook ( ) + O log n n . We proved that the curve ( u ) = ( 2 u sin- 1 u 2 + 2- u 2 | u | 2 , | u | | u | > 2 . minimizes K ( ) among the (rotated) continual Young diagrams. Explain (as rigorously as you can) how this can be used to deduce in a computation-free way the evaluation K () =- 1 2 . 2. Prove the Cauchy determinant identity det 1 x i + y j n i,j =1 = Q 1 i<j<n ( x j- x i )( y j- y i ) Q 1 i,j n ( x i + y j ) . Hint: This determinant can be evaluated in many ways. One of the easiest involves performing a simple elementary operation on columns 2 through n of the Cauchy matrix, extracting factors common to rows and columns and then...
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