$p$-th order generalized Fibonacci cubes and maximal cubes in Fibonacci $p$-cubes

TL;DR

This paper introduces p-th order Fibonacci cubes, generalizing Fibonacci cubes by allowing p consecutive 1s, and explores their combinatorial properties, connections to compositions, and maximal hypercubes.

Contribution

It establishes the structure, enumerative properties, and generating functions of p-th order Fibonacci cubes, and links maximal hypercubes to higher-order Fibonacci cubes.

Findings

Derived formulas for order, size, and cube polynomial of $\

Connected maximal hypercubes in Fibonacci p-cubes to vertices of (p+1)-th order Fibonacci cubes.

Provided generating functions and combinatorial interpretations involving p-nomial coefficients.

Abstract

The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive 1s. We study a one parameter generalization, p-th order Fibonacci cubes $\Gamma^{(p)}_n$, which are subgraphs of $Q_n$ induced by strings without p consecutive 1s. We show the link between vertices of $\Gamma^{(p)}_n$ and compositions of integers with parts in $\{1, 2, \ldots , p\}$. Among other eumerative properties, we study the order, size and cube polynomial of $\Gamma^{(p)}_n$ as well as their generating functions. Many of the given expressions are similar to those for Fibonacci cubes, where the $p$-nomial coefficients play the role of binomial coefficients. We also show that maximal induced hypercubes in Fibonacci $p$-cubes $\Gamma^p_n$ , another generalization of Fibonacci cubes, are connected to vertices of $(p + 1)$-th order Fibonacci cubes. We use this link to determine the maximal cube polynomial of Fibonacci $p$-cubes.