Chains and Antichains in the Bruhat Order for Classes of (0, 1)-Matrices

Abstract

Let \({\mathscr {A}}(R,S)\) denote the set of all matrices of zeros and ones with row sum vector R and column sum vector S. This set can be ordered by a generalization of the usual Bruhat order for permutations. Contrary to the classical Bruhat order on permutations, where permutations can be seen as permutation matrices, the Bruhat order on the class \({\mathscr {A}}(R,S)\) is not, in general, graded, and an interesting problem is the determination of bounds for the maximal length of chains and antichains in this poset. In this survey we aim to provide a self-contained account of the recent developments involving the determination of maximum lengths of chains and antichains in the Bruhat order on some classes of matrices in \({\mathscr {A}}(R,S)\).