Largest part and length of a partition in play

Question

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 $Q_n(t)$?

Answer 1

Your $c_n(\lambda)$ is closely related to what Billey, Konvalinka, Swanson 2020 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 submitted by Swanson, one of the article authors.