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5 Theorem. 5 states that g(n) = u(n) whenever n is not of the special form that appears on the right side. These special numbers are called pentagonal numbers, for a geometric reason. This suggests a possible line of attack for a combinatorial proof: nd a bijection between the partitions of n in an even number of di erent parts and the partitions of n in an odd number of di erent parts. This should work whenever n is not a pentagonal number and a little di culty should arise when n is pentagonal.

N X n =0 N xn = 1 1 x x +1 whenever x 6= 1: Let 1 < x < 1 and consider what happens when N gets larger and larger. 2 Proposition. 2 is known as the geometric series. For example, when x = 1=2 we obtain 1 + = 1 = 2: 1 + 21 + 41 + 81 + 16 1 1=2 45 CHAPTER 7. GENERATING FUNCTIONS 46 In our combinatorial context we are really not interested, at least for the moment, in questions of convergence. Rather we interpret x as a variable and see the geometric series as a formal identity or, if you want, we see the right side 1 as shorthand for the left side.

As we want change for 1 dollar it must be the case that x + 5x + 10x + 25x = 100: The xi must be non-negative integers, and what we want to count is the number of solutions. 4 Problem. 3 is equivalent to the following: Determine the number of solutions of 1 2 1 2 3 3 4 4 x + 5x + 10x + 25x = 100 in non-negative integers xi : It is clear how to generalize this in a natural way: given constants ci; i = 1; : : : r and a right side m; determine the number of solutions of the equation 1 2 3 r X i 4 cixi = m =1 in non-negative integers xi : Recall that we studied the special case when c = = cr = 1 in Chapter 2.

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