Visible Point Vector Partition Identities for Hyperpyramid Lattices

TL;DR

This paper introduces an elementary method to derive identities for visible lattice points in hyperpyramids across multiple dimensions, using gcd-based approaches to study partitions in nD space.

Contribution

It presents a novel gcd-based approach to derive visible point identities in hyperpyramid lattices and introduces new combinatorial identities for these partitions.

Findings

Derived identities for visible points in hyperpyramid lattices.

Established a method to study nD partitions along radial lines.

Proposed new combinatorial identities for vector partitions.

Abstract

We set out an elementary approach to derive Visible Point Identities summed on lattice points of inverted triangle (2D), pyramid (3D), hyperpyramid (4D, 5D and so on) utilizing the greatest common divisor for the nD Visible Point Vectors. This enables study of partitions in nD space into vector parts distributed along straight lines radial from the origin in first hyperquadrant where coordinates of lattice points are all positive integers. We also give several new combinatorial identities for Visible Point Vector partitions.