partitions such that each number $k$ ($1 \le k \le n$) appear atmost $k$ times

Question

Let $\lambda$ be a partition of $n$ such that each number $k$ ($1 \le k \le n$) appear atmost $k$ times in $\lambda$.

For example : $\lambda = 6+6+6+6+3+2+2+1$

is there any special name for these types of partitions?

Also, is it easy to write down the generating function of these partitions like we have for other types of partitions with restrictions?

If possible please share some combinatorial significance of these partitions.

Thanks a lot for your time.

Have a good day.

Answer 1

The generating function is $$\prod_k (1+x^k+x^{2k}+\dots+x^{k^2})=\prod_k\frac{1-x^{k^2+k}}{1-x^k}=\prod_{n\notin\{k^2+k:k=1,2,\dots\}} \frac1{1-x^n},$$ so it is the same as the number of partitions onto summands not belonging to the set $\{k^2+k:k=1,2,\dots\}$.