MATH 145: HOMEWORK 3 ANDREW BERGET This homework is due on Wednesday January 27. In these problem you will need to use the fact that X n ≥0 ax n = a 1-x . Read section 5.3 and solve problems 5.23(a) and 5.25+. Problem A. In this problem we consider the generating function for the Stirling numbers of the second kind, S ( n,k ). Fix k ≥ 0. Deﬁne G k ( x ) = X n ≥0 S ( n,k ) x n (1) Consider the recurrence relation for S ( n,k ): S ( n,k ) = S ( n-1 ,k-1) + kS ( n-1 ,k ) , n,k ≥ 1 . Show this gives rise to a recurrnce relation for G k ( x ): G k ( x ) = xG k-1 ( x ) + kxG k ( x ) . (2) Prove by induction on k that G k ( x ) = x 1-kx x 1-( k-1) x ... x 1-x G0 ( x ) and determine what G0 ( x ) is in order to determine what G k ( x ) is. (3) + Explain why your answer is correct for k = 1. Write down a formula for S ( n, 2)? Problem B. + In how many ways can we distribute 8 balls into 6 boxes if (1) The balls and boxes are all distinct? (2) The balls and boxes are all distinct, box #1 contains 2 balls, box #2 contains 3 balls, box #3
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