# qusol - MATH 239 Quiz (Tuesday) - Suggested Solutions 1. We...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 239 Quiz (Tuesday) - Suggested Solutions 1. We have (2) S ( x ) = s s S x w 2 ( s ) = s s S x aw 1 ( s )+ b = s s S x b ( x a ) w 1 ( s ) = x b s s S ( x a ) w 1 ( s ) = x b (1) S ( x a ) , as required. 2. (a) Let N odd denote the set of odd positive integers, and N even denote the set of even non-negative integers. Let S be the set of ordered lists ( t 1 , . . .t 2 k ) where t 1 , . . ., t k N odd and t k +1 , . . ., t 2 k N even , and let w be given by w ( t 1 , . . .t 2 k ) = t 1 + . . . + t 2 k . Observe that a n is the number of conFgurations in S of weight n , so a n = [ x n ] S ( x ). S = N k odd N k even , and the weight function condition for the Product Lemma is satisFed, so we have S ( x ) = ( N odd ( x )) k ( N even ( x )) k . Now, N odd ( x ) = x + x 3 + x 5 + . . . = x (1+ x 2 + x 4 + . . . ) = x 1-x 2 and N even ( x ) = 1+ x 2 + x 4 + . . . = 1 1-x 2 , so S ( x ) = ( x 1-x 2 ) k ( 1 1-x 2 ) k = x k (1-x 2 ) 2 k , as required. as required....
View Full Document

## qusol - MATH 239 Quiz (Tuesday) - Suggested Solutions 1. We...

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

View Full Document
Ask a homework question - tutors are online