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

Let n,d∈N and n>d. An (n−d)-domino is a box I1×⋯×In such that Ij∈{[0,1],[1,2]} for all j∈N⊂[n] with |N|=d and Ii=[0,2] for every i∈[n]∖N. If A and B are two (n−d)-dominoes such that A∪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∪B=C∪D and {C,D}≠{A,B}, then the pair C,D is called a flip of A,B. A family D of (n−d)-dominoes is a tiling of the box [0,2]n if interiors of every two members of D are disjoint and ⋃B∈DB=[0,2]n. An (n−d)-domino tiling D′ is obtained from an (n−d)-domino tiling D by a flip, if there is a twin pair A,B∈D such that D′=(D∖{A,B})∪{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 D,E of this family the tiling E can be obtained from D by a sequence of flips. In the paper a flip-connected class of (n−d)-domino tilings of the box [0,2]n is described.