[MUSIC] There is a convenient way of representing partitions by means of Young diagrams. Theses are the following pictures. Suppose we have a partition, well, let's take an example, say, 13=5+5+2+1. Let us draw a picture consisting of cells aligned by the left column. And rows of these cells have lengths equal to the entries of the partition. Say in this case we'll have five cells in the first row, five cells in the second row. Two cells and cells are also called boxes. Two boxes in the third row and only one box in the last row. So the rows are left adjusted. And each row consists the same or smaller number of boxes as the previous row. So the second is,well in this example it has the same length as the first one. The third one is shorter than the second one and the fourth one is shorter than the third one. So let us draw all the Young diagrams for n = 4. So let's draw all diagrams consisting of four boxes. So example, for n = 4, we have the following pictures. We have only one row. This is the partition consisting only of one. We have 3 + 1. We have 2 + 2, this is a square. We have, 2 + 1 + 1. And we have a column, Of four boxes. Okay, our next goal will be to say something about the number of partitions of a given number n. Goal. So here are some bad news, that for p(n) there is no explicit formula, as you had for instance for the number of compositions. But in this case we can form the generating function for the sequence p(n) and compute these generating function. And this is what we're going to do in the next part of the lecture. So consider the generating function P(q) which will be equal to the sum p(n) q to the power n for n > 0. So it will look like 1 + q + 2q squared + 3q cubed + 5q to the power of 4 + 7q to the fifth and so on. So our next goal would be to compute this function.