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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.00759 |
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| _version_ | 1866911744843055104 |
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| author | Kisielewicz, Andrzej P. |
| author_facet | Kisielewicz, Andrzej P. |
| contents | Let $n,d\in \mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box $I_1\times \cdots \times I_n$ such that $I_j\in \{[0,1],[1,2]\}$ for all $j\in N\subset [n]$ with $|N|=d$ and $I_i=[0,2]$ for every $i\in [n]\setminus N$. If $A$ and $B$ are two $(n-d)$-dominoes such that $A\cup B$ is an $(n-(d-1))$-domino, then $A,B$ is called a twin pair. If $C,D$ are two $(n-d)$-dominoes which form a twin pair such that $A\cup B=C\cup D$ and $\{C,D\}\neq \{A,B\}$, then the pair $C,D$ is called a flip of $A,B$. A family $\mathscr{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2]^n$ if interiors of every two members of $\mathscr{D}$ are disjoint and $\bigcup_{B\in \mathscr{D}}B=[0,2]^n$. An $(n-d)$-domino tiling $\mathscr{D}'$ is obtained from an $(n-d)$-domino tiling $\mathscr{D}$ by a flip, if there is a twin pair $A,B\in \mathscr{D}$ such that $\mathscr{D}'=(\mathscr{D}\setminus \{A,B\})\cup \{C,D\}$, where $C,D$ is a flip of $A,B$. A family of $(n-d)$-domino tilings of the box $[0,2]^n$ is flip-connected, if for every two members $\mathscr{D},\mathscr{E}$ of this family the tiling $\mathscr{E}$ can be obtained from $\mathscr{D}$ by a sequence of flips. In the paper some flip-connected class of $(n-d)$-domino tilings of the box $[0,2]^n$ is described. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00759 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on a flip-connected class of generalized domino tilings of the box $[0,2]^n$ Kisielewicz, Andrzej P. Combinatorics Metric Geometry Let $n,d\in \mathbb{N}$ and $n>d$. An $(n-d)$-domino is a box $I_1\times \cdots \times I_n$ such that $I_j\in \{[0,1],[1,2]\}$ for all $j\in N\subset [n]$ with $|N|=d$ and $I_i=[0,2]$ for every $i\in [n]\setminus N$. If $A$ and $B$ are two $(n-d)$-dominoes such that $A\cup B$ is an $(n-(d-1))$-domino, then $A,B$ is called a twin pair. If $C,D$ are two $(n-d)$-dominoes which form a twin pair such that $A\cup B=C\cup D$ and $\{C,D\}\neq \{A,B\}$, then the pair $C,D$ is called a flip of $A,B$. A family $\mathscr{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2]^n$ if interiors of every two members of $\mathscr{D}$ are disjoint and $\bigcup_{B\in \mathscr{D}}B=[0,2]^n$. An $(n-d)$-domino tiling $\mathscr{D}'$ is obtained from an $(n-d)$-domino tiling $\mathscr{D}$ by a flip, if there is a twin pair $A,B\in \mathscr{D}$ such that $\mathscr{D}'=(\mathscr{D}\setminus \{A,B\})\cup \{C,D\}$, where $C,D$ is a flip of $A,B$. A family of $(n-d)$-domino tilings of the box $[0,2]^n$ is flip-connected, if for every two members $\mathscr{D},\mathscr{E}$ of this family the tiling $\mathscr{E}$ can be obtained from $\mathscr{D}$ by a sequence of flips. In the paper some flip-connected class of $(n-d)$-domino tilings of the box $[0,2]^n$ is described. |
| title | A note on a flip-connected class of generalized domino tilings of the box $[0,2]^n$ |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2401.00759 |