MATC44 2008 - s4 - University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44 Winter 2008 Solutions to Assignment#4

# MATC44 2008 - s4 - University of Toronto at Scarborough...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C44, Winter 2008 Solutions to Assignment #4 Problem 2. on page 317: Let Σ n be a set of all 2-by- n arrays x 11 x 12 · · · x 1 n x 21 x 22 · · · x 2 n such that x 11 < x 12 < · · · < x 1 n , x 21 < x 22 < · · · < x 2 n , and x 11 < x 21 , x 12 < x 22 , . . . , x 1 n < x 2 n . Let Γ n be a set of all sequences { a k } 2 n k =1 of numbers +1 and - 1 such that a 1 + a 2 + · · · + a k 0, k = 1 , 2 , . . . , 2 n . By Theorem 8.1.1, cardinality of Γ n is equal to n th Catalan number, C n . Therefore, if we establish a bijection between Σ n and Γ n , we’ll have that | Σ n | = C n . Given an 2-by- n array A Σ n , consider a sequences B = { a i } 2 n i =0 such that a i = 1 if i is in the first row of A , and a i = - 1 if i is in the second row of A . We want to show that B Γ n . Indeed, if B is not in Γ, then there exists the smallest k ∈ { 1 , 2 , . . . , n } such that a 1 + a 2 + · · · + a k = - 1. Since k is the smallest such index, it follows that a 1 + a 2 + · · · + a k - 1 = 0, so there is equal number of - 1’s and +1’s and hence k - 1 = 2 j , and a k = - 1. Also this means that there are equally many

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