mat280-hw2-old

mat280-hw2-old - MAT 280 UC Davis Winter 2011 Longest...

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MAT 280 — UC Davis, Winter 2011 Longest increasing subsequences and combinatorial probability Homework Set 2 Homework due: Wednesday 2/23/11 1. If f is a continual Young diagram and g is its rotated version, let K ( g ) = I hook ( f ) = Z 0 Z f ( x ) 0 log h f ( x,y ) dy dx be the hook integral that appears in the asymptotic version of the hook length formula d 2 λ | λ | ! = exp ± - n ² 1 + 2 I hook ( φ λ ) + O ² log n n ³³´ . We proved that the curve Ω( u ) = ( 2 π µ u sin - 1 µ 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 (Ω) = - 1 2 . 2. Prove the Cauchy determinant identity det ² 1 x i + y j ³ 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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This note was uploaded on 11/13/2011 for the course MATH 280 taught by Professor Hunter during the Winter '08 term at UC Davis.

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mat280-hw2-old - MAT 280 UC Davis Winter 2011 Longest...

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