Geodesic
Irreducible subcube partitions
Yuval Filmus1 ; Edward A. Hirsch ; Sascha Kurz ; Ferdinand Ihringer ; Artur Riazanov ; Alexander V. Smal ; Marc Vinyals
1Technion - Israel Institute of Technology
The electronic journal of combinatorics, Tome 30 (2023) no. 3
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A subcube partition is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of $\{0,\dots,q-1\}^n$, and partitions of $\mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.
DOI : 10.37236/11862
Classification : 05D05, 05D99, 51E23
Mots-clés : Hamming weight, irreducibility, Lagarias-Shor construction

Yuval Filmus  1   ; Edward A. Hirsch    ; Sascha Kurz    ; Ferdinand Ihringer    ; Artur Riazanov    ; Alexander V. Smal    ; Marc Vinyals 

1 Technion - Israel Institute of Technology 
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     title = {Irreducible subcube partitions},
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     year = {2023},
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     number = {3},
     doi = {10.37236/11862},
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Yuval Filmus; Edward A. Hirsch; Sascha Kurz; Ferdinand Ihringer; Artur Riazanov; Alexander V. Smal; Marc Vinyals. Irreducible subcube partitions. The electronic journal of combinatorics, Tome 30 (2023) no. 3. doi: 10.37236/11862

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