Lesson 1

Generating functions provide a compact way to encode all one-share portfolios. For the toy market above, consider the polynomial \[F(x)=(1+x^{2})(1+x^{3})(1+x^{5})(1+x^{7}).\] The factor \((1+x^{2})\) corresponds to ALP: choose the term \(1\) if we do not buy it, and choose the term \(x^{2}\) if we do buy it. When we multiply the four factors, choosing one term from each factor is exactly the same thing as choosing a portfolio. Therefore, the coefficient \([x^{10}]F(x)\) is the number of portfolios of total price \(10\).