Structure and enumeration results of matchable Lucas cubes

Abstract

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.