math 435 homework3

math 435 homework3 - α ∈ S n and τ is a transposition...

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Homework 3 for MATH 435 Solutions Problem 1 Book, p. 124, Exercise 2.26 Solution. We know a permutation α is odd i sgn ( α ) = 1. If α is an r -cycle, then by deﬁnition, sgn ( α ) = (- 1 ) n -( n - r )+ 1 = (- 1 ) r + 1 , which is = 1 i r is odd. ± Problem 2 Book, p. 124, Exercise 2.28 Problem 3 Book, p. 124, Exercise 2.34 Problem 4 Book, p. 125, Exercise 2.35 Problem 5 (*) Let n > 1, and let α S n . A pair ( i , j ) , 1 6 i < j 6 n , is an inversion in α if α ( i ) > α ( j ) . Show that sgn ( α ) = ± 1 if # of inversions in α is even, - 1 if # of inversions in α is odd. Solution. The assertion follows from the following observation: If
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Unformatted text preview: α ∈ S n and τ is a transposition, and we let β = τσ , then the di ﬀ erence of the number of inversions of α and β is odd. This can be seen via a case distinction. If τ = ( ij ) , then any constellation between a triple i , j , k can only change either from 0 inversions in α to 3 inversions in β , or from 1 to 2 inversions, or 2 to 1 inversion, or 3 to 0 inversions. This gives an overall count of odd. ±...
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This note was uploaded on 10/13/2010 for the course MATH 435 at Penn State.

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