Partitions with Durfee triangles of fixed size

A well-studied statistic of an integer partition is the size of its Durfee square. In particular, the number $D_k(n)$ of partitions of $n$ with Durfee square of fixed size $k$ has a well-known simple rational generating function. We study the number $R_k(n)$ of partitions of $n$ with Durfee triangle of size $k$ (the largest subpartition with parts $1,2,\dots ,k$). We determine the corresponding generating functions which are rational functions of a similar form. Moreover, we explicitly determine the leading asymptotic of $R_k(n)$, as $n\to \mathrm{\infty }$.