a04 - MAD 6206 Combinatorics I, Fall 2011 CRN: 87401...

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Unformatted text preview: MAD 6206 Combinatorics I, Fall 2011 CRN: 87401 Assignment 4 Question 1 . (Catalan numbers and sequences of “+1”s and “ − 1”s) Assume that p,r are nonnegative integers. Let Ω p,r be the number of sequences { a i } i ≥ 1 , where a i = ± 1, with p “ − 1”s and p + r “+1”s (so that each sequence sums to r , that is, ∑ 2 p + r i =1 a i = r ). Let C p,r = r 2 p + r parenleftbigg 2 p + r p parenrightbigg , where C p, = δ p, . Note that C p, 1 = 1 2 p +1 ( 2 p +1 p ) = 1 p +1 ( 2 p p ) is the (standard) Catalan number defined in class. We showed in class that the generating function C ( z ) = ∑ p ≥ C p, 1 z p of Catalan numbers satisfies C ( z ) = 1 + z C ( z ) 2 . (1) Recall also that C p, 1 equals the number of sequences of p “ − 1”s and p “+1”s so that every partial sum is at most zero. Let f p,r be the in Ω p,r with the property that every partial sum is less than r . (That is, ∑ j i =1 a i < r for all j < 2 p + r .) We shall show that f p,r = C p,r . (a) Prove (using a simple argument, with our knowledge of C p, 1 given in the problem) that f p, 1 = C p, 1 . (b) Prove that f p,r = summationdisplay p 1 + p 2 + ··· + p r = p C p 1 , 1 C p 2 , 1 ··· C p r , 1 , where the sum is over all p i ≥ 0 satisfying p 1 + p 2 + ··· + p r = p ....
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This note was uploaded on 01/02/2012 for the course MAD 6206 taught by Professor Suen during the Spring '11 term at University of South Florida.

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a04 - MAD 6206 Combinatorics I, Fall 2011 CRN: 87401...

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