Intuition behind this q-binomial formula counting sums of subsets

Question

We have these transparent, motivating interpretations for binomial coefficients and their $q$-analogue.

$$ \binom{n}{j} = \begin{matrix} \text{"The number of subsets of size $j$}\\ \text{of a set of size $n$."} \end{matrix} \qquad \newcommand{\qbinom}[2]{\begin{bmatrix}#1\\#2\end{bmatrix}_q} \newcommand{\qbinomsmall}[2]{\left[\begin{smallmatrix}#1\\#2\end{smallmatrix}\right]_q} \qbinom{n}{j} = \begin{matrix} \text{"The number of subspaces of dimension $j$} \\ \text{of a vector space of dimension $n$} \\ \text{over a field with $q$ elements."} \end{matrix} $$

In Quantum Calculus, before talking about this interpretation of the $q$-binomial coefficients, Kac and Cheung prove another formula for the $q$-binomials without motivating it. Let $\mathcal{A}_{n,j}$ denote the set of subsets of size $j$ of the set $\{1,2,\dotsc,n\}$. Then

$$ \qbinom{n}{j} = \sum_{S \in \mathcal{A}_{n,j}} q^{w(S) - \frac{1}{2}j(j+1)} \quad\text{where}\quad w(S) = \sum_{s\in S} s\,. $$

What exactly is this sum counting? I don't immediately see this formula counting vector subspaces. Because it involves $\mathcal{A}_{n,j}$, it will clearly counts subsets of size $j$ of a set of size $n$ in the $q \to 1$ limit. But I don't see the inspiration for this formula. The $\frac{1}{2}j(j+1)$ bit is the minimum value that $w(S)$ can be, so maybe that's just there to normalize the power of $q$? And I don't see the connection, but this formula comes in the book just after the proof that $\qbinomsmall{n}{j}$ is a polynomial in $q$ of degree $j(n-j)$ with symmetric coefficients. So the coefficients of $\qbinomsmall{n}{j}$ will be the number of $S \in \mathcal{A}_{n,j}$ for which $w(S)$ is the same. But I'm still missing why this formula should give you those coefficients.

Answer 1

$ \newcommand{\qbinom}[2]{\begin{bmatrix}#1\\#2\end{bmatrix}_q} \newcommand{\qbinomsmall}[2]{\left[\begin{smallmatrix}#1\\#2\end{smallmatrix}\right]_q} \qbinomsmall{n}{j}$ is the generating function for partitions into at most $j$ parts and parts of size at most $n-j$. (So partitions whose Ferrers diagrams fit within a $n-j$ by $j$ grid) We can represent such partitions as $\pi=(\pi_1,\pi_2,\ldots,\pi_j)$ where $n-j\ge\pi_1\ge\pi_2\ge\cdots\ge\pi_j\ge0$. So $\qbinomsmall{n}{j}$ is the sum of $q^{w(\pi)}=q^{\pi_1+\cdots+\pi_j}$ over all such partitions. Then $(\pi_1',\pi_2',\ldots,\pi_j') =(\pi_1+j,\pi_2+j-1,\ldots,\pi_j+1)$ satisfies $n\ge \pi_1'>\pi_2'>\cdots>\pi_j'\ge1$ so corresponds to a $j$-element subset of $\{1,\ldots,n\}$. Moreover $\pi_1'+\cdots+\pi_j'=w(\pi)+\frac12j(j+1)$.