Lesson 3

If we allow buying several shares of stock \(i\), say from zero to \(m_{i}\) shares, the corresponding factor becomes \[ 1+x^{p_i}y^{r_i}+x^{2p_i}y^{2r_i}+\dotsb+x^{m_i p_i}y^{m_i r_i}. \] One can also introduce a variable \(z\) to count how many different stocks are used: \[ F(x,y,z)=\prod_{i=1}^n\left(1+z x^{p_i}y^{r_i}+z x^{2p_i}y^{2r_i}+\dotsb+z x^{m_i p_i}y^{m_i r_i}\right). \] Then \([x^{b} y^{r} z^{k}]F(x,y,z)\) counts portfolios with price \(b\), score \(r\), and exactly \(k\) different stocks.