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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?

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  • $\begingroup$ What does $|\lambda|$ mean? The number of elements in the partition? $\endgroup$
    – JMP
    Commented May 25, 2023 at 10:11
  • $\begingroup$ It’s the weight of the partition (sun of the parts) $\endgroup$
    – Giulia Lanzafame
    Commented May 25, 2023 at 10:14

1 Answer 1

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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.

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  • $\begingroup$ Ok this is clear but I don’t get how this helps with the equality I’m asking in the question $\endgroup$
    – Giulia Lanzafame
    Commented May 25, 2023 at 10:30
  • $\begingroup$ @guila; the LHS is the q-Pochhammer, and gives a part of size $r$ in the numerator, and any number of $\le$ parts in the denominator. The RHS counts by the weight of each partition in a $P_r$. $\endgroup$
    – JMP
    Commented May 25, 2023 at 10:42
  • $\begingroup$ Ah wow yes this makes it incredibly clear, thanks $\endgroup$
    – Giulia Lanzafame
    Commented May 25, 2023 at 10:46

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