If $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq\lambda\_k)\vdash n$ is an integer partition of $n$ then $\lambda\_1$ is its largest part and $k$ is its length, $\ell(\lambda)$.


Define the statistic $c\_n(\lambda)=\max\{\lambda\_1,\ell(\lambda)\}$ for the above partition. Also consider the polynomial (in $t$),
$$Q\_n(t)=\sum\_{\lambda\vdash n}t^{c\_n(\lambda)}.$$
Here are some examples:



```
                           t

                             2
                          2 t 

                          3    2
                       2 t  + t 

                       4      3    2
                    2 t  + 2 t  + t 

                      5      4      3
                   2 t  + 2 t  + 3 t 

                  6      5      4      3
               2 t  + 2 t  + 4 t  + 3 t 

               7      6      5      4      3
            2 t  + 2 t  + 4 t  + 5 t  + 2 t 

            8      7      6      5      4    3
         2 t  + 2 t  + 4 t  + 6 t  + 7 t  + t 

         9      8      7      6      5      4    3
      2 t  + 2 t  + 4 t  + 6 t  + 9 t  + 6 t  + t 

      10      9      8      7       6       5      4
   2 t   + 2 t  + 4 t  + 6 t  + 10 t  + 11 t  + 7 t 

```

**QUESTION 1.** It appears that the coefficients of $Q\_n(t)$, read from left to right, are twice the partition numbers $1,1,2,3,5,7,11,15,\dots$, up to (at least) the middle term. Is this true?


**QUESTION 2.** Is there a generating function for the polynomials $Q\_n(t)$?



Largest part and length of a partition in play
Your $c\_n(\lambda)$ is closely related to what [Billey, Konvalinka, Swanson 2020](https://arxiv.org/abs/1905.00975) call the aft of a partition:
$$ \text{aft}(\lambda) = | \lambda | - \max\{\lambda\_1, \lambda'\_1\}$$
where $\lambda'$ is the conjugate of $\lambda$, so $\lambda'\_1 = \ell(\lambda)$.


Following up on Martin Rubey's comment, your Question 1 is true: For $c\_n(\lambda) > n/2$, we have $\lambda \ne \lambda'$ since $\lambda\_1 = \lambda'\_1$ would make $2\lambda\_1 > n$, a contradiction. Therefore, $\lambda$ is not self-conjugate and each partition $\mu \vdash (n - \lambda\_1)$ is counted twice in the coefficient of $t^{\lambda\_1}$, once for the partition of $n$ consisting of $\lambda\_1$ followed by $\mu$ and once for the conjugate of that partition of $n$.


For your Question 2, a generating function in terms of Gaussian binomial coefficients is given in the related OEIS entry [A338621](https://oeis.org/A338621) submitted by Swanson, one of the article authors.


