Geodesic
Combinatoric of the karyon tilings
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 54-99

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In this article, we study the combinatorial properties of the karyon tilings $\mathcal{T}$ of the torus $\mathbb{T}^d$ of an arbitrary dimension $d$. Our main results are the following statements: 1) the karyon corona $\mathbf{Cr}$ contains all types of polyhedral stars of the $\mathcal{T}$ tilings; 2) the number of all faces of dimension $a$ of the tiling $\mathcal{T}$ is equal to $md!/((d-a)!a!)$, where $m$ is the order of tilling. 
@article{ZNSL_2022_511_a2,
     author = {V. G. Zhuravlev},
     title = {Combinatoric of the karyon tilings},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {54--99},
     publisher = {mathdoc},
     volume = {511},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a2/}
}
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V. G. Zhuravlev. Combinatoric of the karyon tilings. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 54-99. http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a2/