A note on a flip-connected class of generalized domino tilings of the box $[0,2{]}^n$

Abstract

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 $\{C,D\}\ne \{A,B\}$, then the pair $C,D$ is called a flip of $A,B$. A family $\mathcal{D}$ of $(n-d)$-dominoes is a tiling of the box $[0,2{]}^n$ if interiors of every two members of $\mathcal{D}$ are disjoint and ${\bigcup }_{B\in \mathcal{D}}B=[0,2{]}^n$. An $(n-d)$-domino tiling ${\mathcal{D}}^{\prime }$ is obtained from an $(n-d)$-domino tiling $\mathcal{D}$ by a flip, if there is a twin pair $A,B\in \mathcal{D}$ such that ${\mathcal{D}}^{\prime }=(\mathcal{D}\setminus \{A,B\})\cup \{C,D\}$, 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 $\mathcal{D},\mathcal{E}$ of this family the tiling $\mathcal{E}$ can be obtained from $\mathcal{D}$ 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.