  A large family of nonsingular rational solutions of the
Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions are
constructed via the Gramian method and are identified as points in a complex
Grassmannian. Each solution is a traveling wave moving with a uniform
background velocity but have multiple peaks which evolve at a slower time scale
in the co-moving frame. For large times, these peaks separate and form
well-defined wave patterns in the $xy$-plane. The pattern formation are
described by the roots of well-known polynomials arising in the study of
rational solutions of Painlev\'e II and IV equations. This family of solutions
are shown to be described by the classical Schur functions associated with
partitions of integers and irreducible representations of the symmetric group
of $N$ objects. It is then shown that there exists a one-to-one correspondence
between the KPI rational solutions considered in this article and partitions of
a positive integer $N$.
