On interrelations between strongly, weakly and chord separated set-systems (a geometric approach)
| Autor: | Gleb A. Koshevoy, Vladimir I. Danilov, Alexander V. Karzanov |
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| Rok vydání: | 2021 |
| Předmět: |
Mathematics::Combinatorics
Algebra and Number Theory Algebraic combinatorics 010102 general mathematics Rhombus 0102 computer and information sciences 01 natural sciences Representation theory Set (abstract data type) Combinatorics 010201 computation theory & mathematics Bijection Discrete Mathematics and Combinatorics Chord (music) 0101 mathematics Mathematics |
| Zdroj: | Journal of Algebraic Combinatorics. 54:1299-1327 |
| ISSN: | 1572-9192 0925-9899 |
| DOI: | 10.1007/s10801-021-01047-5 |
| Popis: | We consider three types of set-systems that have interesting applications in algebraic combinatorics and representation theory: maximal collections of the so-called strongly, weakly, and chord separated subsets of a set $$[n]=\{1,2,\ldots ,n\}$$ . These collections are known to admit nice geometric interpretations; namely, they are, respectively, in bijection with rhombus tilings on the zonogon Z(n, 2), combined tilings on Z(n, 2), and fine zonotopal tilings (or “cubillages”) on the 3-dimensional zonotope Z(n, 3). We describe interrelations between these three types of set-systems in $$2^{[n]}$$ , working in terms of their geometric models. In particular, we characterize the sets of rhombus and combined tilings properly embeddable in a fixed 3-dimensional cubillage and give efficient methods of extending a given rhombus or combined tiling to a cubillage, etc. |
| Databáze: | OpenAIRE |
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