  We consider path ideals associated to special classes of posets such as tree
posets and cycles. We express their property of being sequentially
Cohen-Macaulay in terms of the underlying poset. Moreover, monomial ideals,
which arise from the Luce-decomposable model in algebraic statistics, can be
viewed as path ideals of certain posets. We study invariants of these so-called
\emph{Luce-decomposable} monomial ideals for diamond posets and products of
chains. In particular, for these classes of posets, we explicitly compute their
Krull dimension, their projective dimension, their regularity and their Betti
numbers.
