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<inf>k</inf>(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<inf>k</inf>(n) of partitions of n with Durfee triangle of size k (the largest subpartition with parts 1,2,…,k). We determine the corresponding generating functions which are rational functions of a similar form. Moreover, we explicitly determine the leading asymptotic of R<inf>k</inf>(n), as n→∞.