Lpolynominoes

Lpolynominoes are a family of polyominoes shaped like the letter L. They are formed by two perpendicular arms that share a single unit square at their corner. An L-polynomino is specified by a pair of integers (p, q) with p ≥ 2 and q ≥ 2, representing the lengths of the vertical and horizontal arms, respectively. The total number of unit squares is p + q − 1. The standard L-tromino corresponds to (2, 2) and has area 3; the L-tetromino corresponds to (3, 2) with area 4; the L-pentomino corresponds to (3, 3) with area 5, and so on.

Rotations of an L-polynomino yield up to four distinct orientations. Reflections do not produce new shapes up to the common symmetry conventions used in polyomino studies, though some classifications distinguish reflections.

Basic properties include the area n = p + q − 1 and the outer boundary length (perimeter) equal to 2(p + q). L-polynominoes are connected grids of unit squares and form simple examples of non-rectangular polyominoes used in tiling and combinatorial problems.

Counting distinct shapes: for a fixed area n, the number of L-polynomino shapes up to rotation is floor((n − 1)/2). This arises from the constraint p + q = n + 1 with p, q ≥ 2, identifying pairs (p, q) and (q, p) under rotation.

Relation to broader theory: L-polynominoes generalize familiar L-trominoes and appear in tiling puzzles and algorithmic studies of polyomino tilings. They are a straightforward example in discussions of shape enumeration, symmetry, and tiling feasibility within the broader field of polyominoes, introduced through the development of polyomino theory by Golomb.