Equality between q shifted factorials and sum with partitions

Question

I am looking at WilliamY.C.Chen,Qing-HuHou,and Alain Lascoux proof for the Gauss Identity in q shifted factorials. At some point they claim that it is easy to see that

$$ \frac{q^r}{(q ; q)_r}=\sum_{\lambda \in P_r} q^{|\lambda|} $$

Where $ \lambda $ is a partition and $ \ P_r $ is the set of partitions $ \lambda $ with maximal component r.

I really cannot see why this is the case, can anyone help?

Answer 1

This is the classic 'number of partitions with maximum part $k$' equals the 'number of partitions with $k$ parts'.

This can be seen by reading a partition in two ways. For example, $53221$ looks like

*
*
**
****
*****

and if you read the row lengths you get $54211$.

*
**
**
***
*****

with one being the reflection about $y=x$ of the other.