  We introduce a general bipartite-like representation and Schmidt
decomposition of an arbitrary pure state of $N$ indistinguishable fermions,
based on states of $M<N$ and $(N-M)$ fermions. It is directly connected with
the reduced $M$- and $(N-M)$-body density matrices (DMs), which have the same
spectrum in such states. The concept of $M$-body entanglement emerges naturally
in this scenario, generalizing that of one-body entanglement. Rigorous
majorization relations satisfied by the normalized $M$-body DM are then
derived, which imply that the associated entropy will not increase, on average,
under a class of operations which have these DMs as post-measurement states.
Moreover, such entropy is an upper bound to the average bipartite entanglement
entropy generated by a class of operations which map the original state to a
bipartite state of $M$ and $N-M$ distinguishable fermions. Analytic evaluation
of the spectrum of $M$-body DMs in some strongly correlated fermionic states is
also provided.
