[SOUND] [MUSIC] Hi, and welcome back to Introduction to Enumerative Combinatorics. This is lecture seven, it is entitled Partitions. A partition of an integer positive number n is a way of representing it as a sum of positive integers lambda one, lambda two, etc, lambda n. So here is the definition. N = lambda 1 + lambda 2 + etc + lambda K. And if two partitions differ only by the order of summands, they are considered the same. So we can order them, say, in the non-strictly decreasing order. We can suppose that lambda 1 is greater than or equal to lambda 2, greater than or equal to lambda 3, etc greater than or equal to lambda K, and lambda K is at least 1. And we will denote this by the following symbol. n is a partition of lambda, which is lambda 1, etc, lambda K. So here is the difference with compositions. We have dealt with compositions in our previous lectures. And say, they were also presentations of a number as a sum. But two compositions which are different only by the order of summands were considered to be different ones, but they correspond to the same partition. So 3 + 2 and 2 + 3, for example, are different, Compositions, But the same partition. Okay, so let me give you some first examples. Let us write down all the partitions of n for some small n, like one, two, three, four, five. Okay, so if n is equal to 1, there is only one way of representing it as a sum. So it is just 1. So for 2, there are two ways. 2 = 1+1. So the number 3 can be represented as a sum in three ways. 3 = 2 + 1 = 1 + 1 + 1. 4, Has the following presentations. 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. So there are 5 partitions, in this case. And we'll denote the number of partitions of a number n by p of n. The number, Of partitions, Of n is denoted, By p(n). So p(1) = 1. P(2) = 2. P(3) = 3. p(4) = 5. And as an exercise, you can try to show that p(5) = 7. And p(6) + 11. As an exercise, write down all the partitions of 5 and 6. [SOUND] [MUSIC]