mat280-hw2

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online