01m×n

01m×n, typically written as 0-1 m-by-n matrices, denotes the set of binary matrices with m rows and n columns. In such matrices every entry aij belongs to the set {0,1}. These matrices are used to encode incidence or adjacency relations in a compact numeric form.

Interpretations and examples: A 0-1 m×n matrix can represent a relation between two finite sets X and Y where X has m elements and Y has n elements; a 1 entry indicates that the corresponding pair is related. In graph theory, the m×n matrix is the biadjacency matrix of a bipartite graph with parts of sizes m and n. When m = n, a square 0-1 matrix with exactly one 1 in each row and column is called a permutation matrix, representing a bijection.

Properties and counting: The total number of m×n binary matrices is 2^(mn). Row sums and column sums give margins that describe degrees in the corresponding bipartite graph or incidence structure. Matrices with fixed row and column sums are known as contingency tables in statistics; counting the number of such matrices with given margins is a well-studied and computationally challenging problem (often #P-complete). Operations on 0-1 matrices can be performed over the real numbers, or in a Boolean algebra context using logical AND and OR.

Applications and related concepts: 0-1 matrices appear in statistics (contingency tables), computer science (sparse relations, data encoding), and image processing (binary images). Related notions include Ferrers matrices with monotone row/column properties and permutation matrices as a special square case. See also 0-1 matrix, adjacency matrix, and incidence matrix.