Euler's Generating Function for the Partition Numbers

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 .