lecture08 - Great Theoretical Ideas In Computer Science A...

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9/22/2011 1 Counting III: Catalan numbers, Generating functions Great Theoretical Ideas In Computer Science A. Gupta V. Guruswami CS 15-251 Fall 2011 Lecture 8 September 22, 2011 Carnegie Mellon University 1 (1-X) n = k = 0 X k k+n-1 n-1 m x x x r 2 1 . . Let us recap some key results from last lecture How many integer nonnegative solutions to the following equations? m x x x r 2 1 ... m 1 r m 1 - r 1 - r m The Binomial Formula k n k n 0 k n y x k n y) (x The Multinomial Formula    i k ,, r=n 3k 12 2 1 k n 2 1 r r r r 3 2 1 k r r . ..,r 2 1 k X +X +. ..+ X n = X X X . ..X r ;r ;. ..;r n r 1 ;r 2 ;...;r k 0, if r 1 r 2 ... r k n n! r 1 !r 2 !...r k ! Manhattan walk All the avenues numbered 0 through x , run north-south, and all the streets, numbered 0 through y, run east-west. The number of [sensible] ways to walk from the corner of 0th st. and 0th avenue to the opposite corner of the city equals: 0 x (0,0) (x,y) xy y  
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9/22/2011 2 level n (n total steps) k’th Avenue 1 1 1 1 1 1 1 1 1 1 1 3 2 3 4 4 6 15 6 6 5 5 10 10 15 20 Manhattan walk and Pascal’s traingle k n 1 1 = # Manhattan walks to this node n-k . . . . . . . . . . . . . . . . . . . . . . . . . level n (n total steps) k’th Avenue 1 1 1 1 1 1 1 1 1 1 1 3 2 3 4 4 6 15 6 6 5 5 10 10 15 20 Pascal’s identity via Manhattan walks 1 - k 1 - n k 1 - n k n 1 1 Level n k th avenue 0 1 2 3 4 n k 2 How many ways to get to via ? Level 2n k th avenue 0 1 2 3 4 2n n n k k = 0 n 2 = End Recap End recap What if we require the Manhattan walk to never cross the diagonal ? How many ways can we walk from (0,0) to (n,n) along the grid subject to this rule? Noncrossing Manhattan walk n n (n,n) (0,0)
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9/22/2011 3 14 such walks for n=4 (c.f. total # Manhattan walks = = 70 ) 4 8 Let’s count # violating paths, that do cross the diagonal Will do so by a bijection. Find first step above the diagonal. “Flip” the portion of the path after that step. Flip the portion of the path the first edge above the diagonal. Note: New path goes to (n-1,n+1) Claim (think about it): Every Manhattan walk from (0,0) to (n-1,n+1) can be obtained in this fashion in exactly one way 1 2 2 n n n n Thus, number of noncrossing Manhattan walks on n x n grid = How many sequences of n 0’s and n 1’s are there such that every prefix has more 0’s than 1’s? The above is the n’th Catalan number.
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lecture08 - Great Theoretical Ideas In Computer Science A...

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