denomino

Denomino is a term used in recreational mathematics to describe a family of tiling problems in which standard domino tiles (2x1 rectangles) are augmented with a positive integer label, called a denominator, assigned to each tile. An instance consists of a rectangular region on the square lattice, a prescribed multiset of denominators, and a target value or set of target values for row sums and column sums. A legal denomino tiling both covers the region with dominoes and assigns the denominators to tiles so that the sum of denominators in every row and every column matches the respective target.

Variants of denomino include uniform denomino, where all rows and columns share the same target, and irregular denomino, where targets vary by row or column. Some formulations require the row sums to form a given sequence, while others require that the multiset of denominators be used exactly once or according to specified multiplicities. In some presentations, a denomino instance may also fix the exact placement of denominators or allow them to be chosen freely under the sum constraints.

Purpose and study: Denomino is studied as a combinatorial labeling problem that combines elements of domino tilings and integer partitioning. It generalizes standard domino tilings by adding arithmetic constraints, and specific cases reduce to ordinary domino tilings when all denominators equal 1.

Computational aspects: Finding a valid denomino tiling is a constraint satisfaction problem; counting distinct denomino tilings for a region is related to enumeration of matchings with additional labeling constraints and can be computationally challenging. Methods used include backtracking, transfer-matrix techniques, and graph-theoretic approaches.

See also: Domino tiling; Tiling problem; Graph matching; Transfer-matrix method; Combinatorial enumeration.