This is a follow up on [my earlier MO question][0].

Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its [conjugate partition][1] $\lambda'$. Denote by $\lambda''=\lambda',0$ found by appending one extra zero at the right end of $\lambda'$. Further, define the following two numerics $a(\lambda'')_j=\lambda_j''-\lambda_{j+1}''$ for $j=1,2,\dots,\ell(\lambda')$.

For example, if $\lambda=(4,2,1)$ then $\lambda'=(3,2,1,1)$ and $\lambda''=(3,2,1,1,0)$ and $a(\lambda'')=(1,1,0,1)$.

>**QUESTION.** Is it true that the coefficients of the polynomial $A_n(q)$ are all in $\{-1,0,1,2\}$? 
$$A_n(q):=\sum_{\lambda\vdash n}q^{n-\ell(\lambda)}
\prod_{a(\lambda'')_j\geq1}\frac{(q^{2a(\lambda'')_j}-1)(q-1)}{q+1}.$$

**REMARK.** In fact, it appears that only coefficient of the middle-term can possibly be equal to $2$.



[0]: https://mathoverflow.net/questions/353715/an-identity-for-polynomials-over-partitions
[1]: http://mathworld.wolfram.com/ConjugatePartition.html