[SOUND] Sometimes it's more convenient to use another form of the previous equality. Namely, proposition, the key binomial coefficient m plus n chose n is equal to the the following fraction 1 minus q to the power of m plus 1. Times (1-q^m+2) times et cetera, until (1-q^m+n) So this expression consists of n factors / (1- q) times (1- q squared) times (1- q to the power n. So, the number of factors in the denominator is the same, which is n. Indeed, let us prove this. As we know M+n choose n is equal to [m+n]! Divided by, [m] and times n factorial. So, we can divide the numerator and denominator by n factorial and get n factorial in the denominator. And m plus one times m plus two times ect, times m plus n in the numerator. So, here we are. M+1 times m + 2 times ect times m + n. So, we know an explicit formula for each of these q integers. It is- 1- q to the power M + 1 divided by 1- q x 1- q to the power m+2 divided by 1-q x etc x 1- q to the power of m plus n divided by 1- q and this is the numerator and the denominator is 1- q divided by 1- q times 1- q squared divided by 1- q etc. 1- q to the power n divided by 1- q. And the number of multiples is the same in the enumerator and the denominator, so we get that this is equal to 1- q to the power m + 1, etc, times 1- q to the power m + n divided by 1- q times etc, times 1- q to the power n. The proposition is proved, so what happens if now m turns to infinity. Suppose that m is very large, let's say 1 million. That means that- This expression in the numerator is almost 1. The first term, well let's say the first million terms of this expression will be zeroes. And this means that the limit of the numerator will be one as m tends to infinity. And the limit of the, and the denominator does not depend upon m, so the limit of this expression, m + n choose n as m goes to infinity is equal to 1 over 1- q times 1- q squared, times etc times 1- q to the power n and this is exactly the expression for which we have seen in our previous lecture, this is the gen reading function which is called Pn(q). [INAUDIBLE] Partitions such that the length of each row does not exceed n. So this is the generating function for partitions with parts not exceeding n. So, in a sense these are, Young diagrams, Of fixed width but, Of arbitrary height. So, they are located inside a rectangle of size n times infinity. We have obtained another proof of this equality. And what happens if both m and n tend to infinity? Then, the limit, Of the expression for the q binomial coefficient m + n choose n. Well of course it's not a polynomial anymore, it's a series. This is our familiar FN product 1 over 1- q to the power of k rrom one to infinity. And this is exactly others generating function or partitions. [SOUND] [MUSIC]