Two mathematical aspects of the centuries-old Japanese sashiko stitching form hitomezashi are discussed: encoding designs using words from a binary alphabet, and duality. Traditional analysed these two ideas. Self-dual related to Fibonacci snowflakes, which we term Pell persimmon polyomino patterns, proposed. Both used generate them appear be new their respective literatures.

When can a polyomino piece of paper be folded into unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for one or several to fold cube, and under which cube folding is impossible. In particular, we show that all five special “basic” guarantee foldability.

We calculate the generating functions for the number of tilings of rectangles of various widths by the right tromino, the L tetromino, and the T tetromino. This allows us to place lower bounds on the entropy of tilings of the plane by each of these. For the T tetromino, we also derive a lower bound from the solution of the Ising model in two dimensions.

The number of line-convex directed polyominoes with given horizontal and vertical projections is studied. It is proven that diagonally convex directed polyominoes are uniquely determined by their orthogonal projections. The proof of this result is algorithmical. As a counterpart, we show that ambiguity can be exponential if antidiagonal convexity is assumed about the polyomino. Then, the result...

We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. Fo...

A permutomino of size n is a polyomino whose vertices define a pair of distinct permutations of length n. In this paper we treat various classes of convex permutominoes, including the parallelogram, the directed convex and the stack ones. Using bijective techniques we provide enumeration for each of these classes according to the size, and characterize the permutations which are associated with...