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A filter lattice is a distributive lattice formed by all filters of a posetin the anti-inclusion order. We study the combinatorial properties of the Hassediagrams of filter lattices of certain posets, so called Fibonacci-like cubes,in this paper. Several enumerative polynomials, e.g.\ rank generating function,cube polynomials and degree sequence polynomials are obtained. Some of theseresults relate to Fibonacci sequence and Padovan sequence.

Some enumerative properties of a class of Fibonacci-like cubes