Combinatorial Matrices

Matrices contain combinatorial information. They may provide alternative representations of combinatorial ideas. Examples include permutation matrices as representations of permutations of a finite set, and adjacency matrices as representations of a finite graph. The linear algebraic properties of these matrices may provide useful combinatorial information, and combinatorial information about a matrix may impact its linear algebraic properties. At the same time, some combinatorial constructs are defined by matrices. A notable example is the alternating sign matrices which arise in a number of ways including from the partial order on permutations called the Bruhat order. In this talk we will explore various connections between combinatorics and matrices, combinatorial matrices.