2019/07/05 by Shuxing Li, Alexander Pott, Li, Shuxing +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG
paper · pdf · doi:10.48550/arxiv.1907.04208
arXiv admin note: substantial text overlap with arXiv:1810.05433. This version contains some minor corrections to version 1
arxiv created 2019/07/12 · arxiv updated 2019/07/15
The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in Li and Pott (arXiv:1810.05433v3), we have little information about primitive formally dual pairs having subsets with unequal sizes. In this paper, we propose a direct construction of primitive formally dual pairs having subsets with unequal sizes in ℤ2 × ℤ42m, where m ≥ 1. This construction recovers an infinite family obtained in Li and Pott (arXiv:1810.05433v3), which was derived by employing a recursive approach. Although the resulting infinite family was known before, the idea of the direct construction is new and provides more insights which were not known from the recursive approach.