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

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.

97 pages, 3 figures, to be submitted as several chapters to be published in the book "Vector Partitions Visible Points and Ramanujan Functions" (Taylor & Francis 2023)