标题： 向量分拆恒等式用于$2$D，$3$D 和$n$D 格子 显示英文标题

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

摘要： 我们证明生成高维向量分拆的恒等式。 我们推导了二维第一象限中整数格点的定理，然后将该方法推广以找到三维和$n$空间格点向量区域扩展。 我们还陈述了二维到五维可见点向量的组合恒等式以及$n$维第一超象限和超金字塔格点。 还推导了具有二元分量的向量分拆的二维和三维定理。 显示英文摘要

摘要： 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.