Point Counts of D_k and Some A_k and E_k Integer Lattices Inside Hypercubes

Abstract

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm in the host lattice. The results assume polynomial format as a function of the hypercube edge length.

Merged chapters 9 and 10, and added Table 5 and Conjecture 2.

References in corpus (3)

• Geometry of tropical moduli spaces and linkage of graphs
• Connection between ordinary multinomials, generalized Fibonacci numbers, partial Bell partition polynomials and convolution powers of discrete uniform distribution
• Motzkin numbers, central trinomial coefficients and hybrid polynomials