# Some enumerative properties of a class of Fibonacci-like cubes

## TL;DR

This paper investigates the combinatorial properties of Fibonacci-like cubes, a class of posets, deriving enumerative polynomials and revealing connections to Fibonacci and Padovan sequences.

## Contribution

It introduces new enumerative polynomials for Fibonacci-like cubes and explores their relationships with classical sequences, expanding understanding of their combinatorial structure.

## Key findings

- Derived rank generating functions for Fibonacci-like cubes
- Established connections between these polynomials and Fibonacci and Padovan sequences
- Provided explicit formulas for degree sequence polynomials

## Abstract

A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams 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 these results relate to Fibonacci sequence and Padovan sequence.