  Entanglement entropy under a particle bipartition provides complementary
information to mode entanglement as it is sensitive to interactions and
particle statistics at leading order and does not depend on any externally
imposed length scale. In this paper, we investigate the particle entanglement
entropy in a system of $N$ interacting spinless lattice fermions in one spatial
dimension by combining bosonization techniques with exact and approximate
numerical methods. We introduce a general scaling form for the fermionic
particle entanglement entropy captured by a shape function that enters as a
extensive interaction induced correction to a known free fermion result. A
general asymptotic expansion in the total number of particles demonstrates that
its form is robust for different values of the R\'enyi index and highlights how
quantum correlations are encoded in the $n$-particle density matrix of a pure
many-body quantum state.
