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This Demonstration shows the partition numbers  up to  as the coefficients of  in the polynomial on the right. The partition numbers  are 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, …. They count the number of ways of splitting an integer  into a sum of positive integers, without regard to order. For instance,  , the number of ways to partition 4, is 5 because there are 5 possible sums that add to 4: 4, 1+3, 2+2, 1+1+2, 1+1+1+1. By convention,  . The left-hand side of the identity shows the product of geometric series  with common ratios  ,  ,  ,…,  , each truncated to  terms. The right-hand side multiplies them out and collects terms. As an example, let  ; the term  on the right is the sum of the five products  ,  ,  ,  ,  , where the factors in each product are in the order of the series they come from. These products correspond to the partitions 4, 1+3, 2+2, 1+1+2, 1+1+1+1. In general, an uncollected term in the expansion of the left side is of the form  . The factor  comes from the  term of the  truncated series  and contributes  (  times) in the corresponding partition. Collecting terms amounts to counting the number of ways  can be partitioned. The generating function for  is thus  . |
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