A1b - P = x 1 1-P + ( u-1) P , for P P ( x, u ) , where x...

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DUE FRIDAY, 1 OCTOBER AT 10:31PM (1) (20 points) Give an expression for [ x n ] G as an explicit function of n of each of the following two functional equations. (a) (10 points) G = F, where F satisfies the functional equation F = x (1 - F ) m , where m is a positive integer. (b) (10 points) G = e T where T satisfies the functional equation T = xe T . (2) (20 points) A regular n -gon (that is, a regular polygon with n sides) A in the plane has a distinguished edge. A diagonal of A is a line segment, in the interior of A , joining two vertices of A. Find the number, c n , of dissections of A into triangles by diagonals that meet only at vertices of A. (3) (20 points) Find the number, a n , of plane planted trees on n non-root vertices with no vertices of degree 2 . Your answer should give a n as an explicit function of n. (4) (20 points) (a) (10 points) Solve the functional equation
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Unformatted text preview: P = x 1 1-P + ( u-1) P , for P P ( x, u ) , where x and u are indeterminates, by giving x n u k P ( x, u ) as an explicit expression in n and k. (b) (10 points) Construct a set P of combinatorial structures and a weight function for P such that [( P , )] o = P ( x, u ) . Note that the weight function must be of the form : P { , 1 , 2 , . . . } 2 : 7 ( 1 ( ) , 2 ( )) where 1 , 2 : P { , 1 , 2 , . . . } . and the form of the generating series P is P ( x, u ) = X P x 1 ( ) u 2 ( ) . ( Remark: You have been introduced to bivariate generating series (generating series in two indeterminates) in Math 239. ) 1...
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This note was uploaded on 11/18/2010 for the course CO 330 taught by Professor R.metzger during the Spring '05 term at Waterloo.

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