Let $\lambda\vdash n$ denote an [integer partition](https://en.wikipedia.org/wiki/Partition_(number_theory)) of $n$ and $\frak{H}\_{\lambda}$ be the multiset of [hook lengths](https://en.wikipedia.org/wiki/Hook_length_formula) of $\lambda$. Further, let $o(\lambda)=\#$ of odd entries and $e(\lambda)=\#$ of even entries in $\frak{H}\_{\lambda}$.


**Example.** If $\lambda=(4,3,1,1)$ then $\frak{H}\_{\lambda}=(7,4,3,1,5,2,1,2,1)$. Also, $o(\lambda)=6$ and $e(\lambda)=3$.



> 
> **QUESTION.** Is there a generating function for the following "parity distribution" function?
> $$F\_n(q,t):=\sum\_{\lambda\vdash n}q^{o(\lambda)}t^{e(\lambda)}.$$
> 
> 
> 


**Examples.** Here is a short list of such polynomials:
\begin{align\*}
F\_1(q,t)&=q, \qquad \qquad F\_2(q,t)=2qt, \qquad F\_3(q,t)=q^3+2q^2t, \\
F\_4(q,t)&=5q^2t^2, \qquad F\_5(q,t)=2q^4t+5q^3t^2.
\end{align\*}


**Remark.** Observe that $\sum\_{n\geq0}F\_n(q,q)\,x^n=\prod\_{i\geq1}\frac1{1-q^ix^i}$.


**Remark.** The coefficient of $q^nt^n$ in $F\_{2n}(q,t)$ is the convolution of $\sum\_{k=0}^np(k)p(n-k)$ where $p(n)$ are the [partitions numbers](https://en.wikipedia.org/wiki/Partition_(number_theory)).



Generating function for parity in hooks
Yes, the generating function is
$$\sum\_{n\geq 0} F\_{n}(q,t)x^n=\prod\_{k\geq 1}\frac{(1-q^{2k}x^{2k})^2}{1-q^kx^k}\cdot \prod \_{k\geq 1}\frac{1}{(1-q^kt^kx^{2k})^2}.$$
This follows from the usual bijection between partitions and 2-cores (corresponding to the first product on the right) and 2-quotients (corresponding to the second product) upon realizing that the boxes in a partition with even hook length are in bijection with the boxes of its 2-quotient.


Incidentally these polynomials are also homogenized versions of the Poincare polynomials of G-Hilbert schemes for $G=\mathbb Z/2\mathbb Z$. For example, see Theorem 2 of "On generating series of classes of equivariant Hilbert schemes of fat points", by S.M. Gusein-Zade, I. Luengo, A. Melle Hernandez, [arXiv:0905.1779](https://arxiv.org/abs/0905.1779).


