Vector Partition Identities for $2$D, $3$D and $n$D Lattices

TL;DR

This paper develops new mathematical identities for vector partitions across various dimensions, extending classical lattice point counting methods to higher dimensions and exploring combinatorial properties of visible point vectors.

Contribution

It introduces novel identities for vector partitions in 2D, 3D, and nD lattices, generalizing previous results and including combinatorial identities for visible point vectors.

Findings

Derived identities for 2D lattice points in the first quadrant

Extended theorems to 3D and nD lattice point regions

Established combinatorial identities for visible point vectors up to 5D

Abstract

We prove identities generating higher dimensional vector partitions. We derive theorems for integer lattice points in the 2D first quadrant, then generalize the approach to find 3D and $n$-space lattice point vector region extensions. We also state combinatorial identities for Visible Point Vectors in 2D up to 5D and $n$D first hyperquadrant and hyperpyramid lattices. 2D and 3D theorems for vector partitions with binary components are also derived.