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 $Q_n(t)$?
