Independent subsets of powers of paths, and Fibonacci cubes

TL;DR

This paper derives a formula for counting edges in the Hasse diagram of independent subsets of powers of paths, revealing a convolution structure involving Fibonacci-like sequences, generalizing Fibonacci cubes.

Contribution

It introduces a general formula for the number of edges in the Hasse diagram of independent subsets of path powers, extending Fibonacci cube properties.

Findings

Number of edges is given by convolution of Fibonacci-like sequences.

Special case h=1 corresponds to Fibonacci cubes.

Provides a unified formula for all powers of paths.

Abstract

We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself.