Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I recently stumbled across the two seemingly similar identities $$ \prod_{i\geq 1}\frac{1}{1-xq^i}=\sum_{n\geq 0}\frac{x^nq^n}{(1-q)(1-q^2)\cdots(1-q^n)} $$ and $$ \prod_{i\geq 1}(1+xq^i)=\sum_{n\geq 0}\frac{x^nq^{\binom{n+1}{2}}}{(1-q)(1-q^2)\cdots(1-q^n)}. $$

Out of curiosity, is there some combinatorial interpretation for these identities, to understand why they hold? Thanks.

If this would better be asked as two questions, I am glad to split it into two.

share|improve this question
add comment

1 Answer 1

up vote 10 down vote accepted

The LHS of the first identity is a generating function $$\prod_{i \ge 1} \frac{1}{1 - xq^i} = \sum p_{m,n} q^m x^n$$

where $p_{m,n}$ counts the number of ways to partition the number $m$ into a sum of $n$ positive integers. In other words, $p_{m,n}$ counts the number of Ferrers diagrams with $m$ dots and $n$ rows.

For fixed $n$, given such a Ferrers diagram slice off the leftmost column. The remaining columns form a partition into parts of size at most $n$, and such partitions have generating function $\frac{1}{(1 - q)...(1 - q^n)}$. The leftmost column contributes a factor of $q^n$, and the fact that we started with a Ferrers diagram with $n$ rows contributes a factor of $x^n$. This gives the RHS of the first identity.


The LHS of the second identity counts the number of ways to partition the number $m$ into a sum of $n$ distinct positive integers. To get the RHS, instead of slicing off the leftmost column of the corresponding Ferrers diagram, you can slice off a right triangle with side lengths $n$ and $n$ because the number of dots in each row are distinct (draw a diagram to convince yourself of this). This triangle has ${n+1 \choose 2}$ dots and the rest of the argument is the same as above.

share|improve this answer
    
+1: I knew of the first interpretation, but the second was new to me. Any reference? Sagan or Stanely, perhaps? –  Alex Youcis Dec 18 '11 at 22:33
    
I wouldn't be surprised if the second was in Stanley somewhere but I think I first saw it in a course that didn't work from a textbook. –  Qiaochu Yuan Dec 18 '11 at 22:53
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.