1x2n

1x2n is a notation used in mathematics to describe a rectangle with height 1 and width 2n units on a square lattice. It is common in tiling problems, grid combinatorics, and as a basic example in discussions of rectangular dimensions.

In tiling contexts, the 1×2n rectangle is typically considered with dominoes of size 1×2 (or 2×1) aligned to the grid. Because the rectangle has height 1, only horizontal placements are possible, and the entire length must be filled by n adjacent dominoes. Consequently, for any positive integer n, there is exactly one tiling of a 1×2n rectangle with such dominoes. More generally, the number of ways to tile a 1×m rectangle with 1×2 dominoes is 1 if m is even and 0 if m is odd.

Beyond tiling, the notation 1×2n also appears in linear algebra and matrix theory. It can denote a row vector or a 1-by-2n matrix, consisting of 2n entries. Its transpose yields a 2n-by-1 column vector. In geometry, a 1×2n rectangle has area 2n square units.

Summary: 1x2n primarily denotes a 1-by-2n rectangle. In domino tilings, it has a single valid tiling for every positive n. The term also functions as a standard dimension or shape descriptor in linear algebra and geometry, where its interpretation depends on context. See also domino tiling, row vectors, and matrix dimensions.