  A lucasene is a hexagon chain that is similar to a fibonaccene, an $L$-fence
is a poset the Hasse diagram of which is isomorphic to the directed inner dual
graph of the corresponding lucasene. A new class of cubes, which named after
matchable Lucas cubes according to the number of its vertices (or elements),
are a series of directed or undirected Hasse diagrams of filter lattices of
$L$-fences. The basic properties and several classes of polynomials, e.g.\ rank
generating functions, cube polynomials and degree sequence polynomials, of
matchable Lucas cubes are obtained. Some special conclusions on binomial
coefficients and Lucas triangle are given.
