If you maximize $P(n,k)$ over $k$ you obtain the OEIS sequence A000792, which appears in a variety of contexts (maximal size of an Abelian subgroup of the symmetric group $S_n$), maximum number of maximal cliques possible in a graph with $n$ vertices, largest number of complexity $n$, ...)

If you maximize $P(n,k)$ over $k$ you obtain the OEIS sequence A000792, which appears in a variety of contexts (maximal size of an Abelian subgroup of the symmetric group $S_n$, maximum number of maximal cliques possible in a graph with $n$ vertices, largest number of complexity $n$, ...)