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Start with some notations: $(a,q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, shortened by $(a)_n$, and $(a)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k)$.

It's easy to verify (using algebraic means) that, for each $m\in\mathbb{Z_{\geq0}}$, $$\sum_{n\geq0}\frac{q^{n^2}q^{mn}}{(q)_n(q)_{n+m}}=\frac1{(q)_{\infty}}. \tag1$$

QUESTION. What is a combinatorial (conceptual) reason for (1) to be independent of $m$?

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  • $\begingroup$ I'm actually not so sure how to interpret this equation when $m$ is negative. Can you clarify what $(q)_{-5}$ means? $\endgroup$ – Sam Hopkins Jun 3 '20 at 16:27
  • $\begingroup$ Perhaps, I'll drop $m<0$. $\endgroup$ – T. Amdeberhan Jun 3 '20 at 17:37
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The RHS is the (size) generating function for all integer partitions.

The LHS is a modification of the idea of keeping track of the Durfee square of a partition.

Namely, for $m\in\mathbb{N}$, let us define for a partition $\lambda$ the $m$-Durfee square of $\lambda$ to be the $n\times (n+m)$ rectangle of boxes in the upper left corner of $\lambda$, where $n$ is as large as it can be. (Note we can have $n=0$, so every partition has such an $m$-Durfee square.) Maybe it should be called "Durfee rectangle."

Then $\frac{q^{n^2}q^{mn}}{(q)_n(q)_{n+m}}$ is easilly seen to be the generating function of partitions with an $m$-Durfee square of dimension $n\times (n+m)$. Summing over all $n$ gives the generating function of all partitions.

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  • $\begingroup$ Is it so common that its shape reminds you the combinatorial objects behind, or there's other way to see why that's the case? $\endgroup$ – Student Jun 3 '20 at 18:58
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    $\begingroup$ Are you asking about the last paragraph? The generating function for partitions by size of Durfee square is very well known: we form any partition placing a $n\times n$ Durfee square, then putting a partition with at most $n$ columns below it, and a partition with at most $n$ rows to its right. For $m$-Durfee squares the reasoning is exactly the same. $\endgroup$ – Sam Hopkins Jun 3 '20 at 19:04

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