Eg x 1 1 x 1 x 2 1 x 1 x 2 x 3 x 1 x 2 x 3 a 2 let

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eg (x 1 ) 1 (x 1 x 2 ) 1 ((x 1 x 2 ) x 3) (x 1 (x 2 x 3 )) A 2 Let the number be b n . Then b 1 = 1, b 2 = 1, b 3 = 2. To bracket n letters, bracket first r, last n - r b n = , n $2 j n & 1 r 1 b r b n & r Let b 0 = 0. Then b n = , n$ 2 j n r 0 b r b n & r Let B(x) = j n $0 b n x n (B(x)) 2 = , c n = j n$ 0 c n x n j n r $0 b r b n & r = j n$ 2 b n x n

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S. Tanny MAT 344 Spring 1999 89 = B(x) - x ² B(x) 2 - B(x) + x = 0. B(x) = 1 & 4x 2 Two possible solutions - must check each = 1 & 4x j n $0 ½ n & 4 n x n Show ½ n & 4 n ’ & 2 n 2n & 2 n & 1 , n$ 1 = 1 2 1 & 4x 1 2 & j n $1 1 n 2n & 2 n & 1 x n ² B(x) = (-ive root req’d!!) j n$ 1 1 n 2n & 2 n & 1 x n ² b n = Catalan (1814-94) 1 n 2n & 2 n & 1 If we take the positive root we get B(x) = 1 - j n $1 1 n 2n & 2 n & 1 x n which give only negative values for the coefficients for n$ 1, which makes no sense. A variety of problems lead to essentially the same recurrence as the above one: (1) counting the number of simple, ordered rooted (SOR) trees - unlabeled rooted trees, each vertex has 0, 1, or 2 descendents, “left” and “right” descendents distinguished
S. Tanny MAT 344 Spring 1999 90 Root Root (2) Secondary structure in RNA [not precisely but similar - see Roberts] (3) Triangulation of an n-gon by diagonals - division of the inside into triangles using only non- intersecting diagonals (4) Let S n be the number of distinct ordered sets of n integers a 1 , a 2 , ³ , a n (allow some to be 0) such that a 1 + ³ + a n = n, a 1 + a 2 + ³ + a k $k for each k < n. Then S n = 1 n % 1 2n n (5) Let S n be the no. of sequences of length 2n, a 1 ,a 2 , ³ , a 2n , a i = + 1 or ! 1 and j 2n j 1 a j 0 , j k j 1 a j$ 0 , k < 2n Then S n = 1 n % 1 2n n INCLUSION - EXCLUSION How many positive integers between 1 and 30 are not divisible by 2 or 3?

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S. Tanny MAT 344 Spring 1999 91 6 = 2 × 3.
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• Fall '06
• miller
• Negative and non-negative numbers, Pallavolo Modena, Sisley Volley Treviso, Fabrique Nationale de Herstal, S. Tanny

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