Gray codes for column-convex polyominoes and a new class of distributive lattices

We introduce the problem of polyomino Gray codes, which is the listing of all members of certain classes of polyominoes such 4 that successive polyominoes differ by some well-defined closeness condition (e.g., the movement of one cell). We discuss various 5 closeness conditions and provide several Gray codes for the class of column-convex polyominoes with a fixed number of cells 6 in each column. For one of our closeness conditions, a natural new class of distributive lattice arises: the partial order is defined 7 on the set of m-tuples [S1] × [S2] × · · · × [Sm ], where each Si > 1 and [Si ] = {0, 1, . . . , Si − 1}, and the cover relations are 8 (p1, p2, . . . , pm) ≺ (p1 + 1, p2, . . . , pm) and (p1, p2, . . . , p j , p j+1, . . . , pm) ≺ (p1, p2, . . . , p j − 1, p j+1 + 1, . . . , pm). We 9 also discuss some properties of this lattice. 10 c © 2007 Elsevier B.V. All rights reserved. 11