mat280-hw2

# mat280-hw2 - MAT 280 — UC Davis Winter 2011 Longest...

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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 λ | λ | ! = exp- n 1 + 2 I hook ( φ λ ) + O log n √ n . We proved that the curve Ω( u ) = ( 2 π 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 (Ω) =- 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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mat280-hw2 - MAT 280 — UC Davis Winter 2011 Longest...

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