  The existence and nature of tripartite entanglement of a noninteracting Fermi
gas (NIFG) is investigated. Three new classes of parameterized entanglement
witnesses (EWs) are introduced with the aim of detecting genuine tripartite
entanglement in the three-body reduced density matrix and discriminating
between the presence of the two types of genuine tripartite entanglement, W\B
and GHZ\W. By choosing appropriate EW operators, the problem of finding GHZ and
W EWs is reduced to linear programming. Specifically, we devise new W EWs based
on a spin-chain model with periodic boundary conditions, and we construct a
class of parametrized GHZ EWs by linearly combining projection operators
corresponding to all the different state-vector types arising for a
three-fermion system. A third class of EWs is provided by a GHZ stabilizer
operator capable of distinguishing W\B from GHZ\B entanglement, which is not
possible with $W$ EWs. Implementing these classes of EWs, it is found that all
states containing genuine tripartite entanglement are of W type, and hence
states containing GHZ\W genuine tripartite entanglement do not arise. Some
genuine tripartite entangled states that have a positive partial transpose
(PPT) with respect to some bipartition are detected. Finally, it is
demonstrated that a NIFG does not exhibit "pure" W\B genuine tripartite
entanglement: three-party entanglement without any separable or biseparable
admixture does not occur.
