Antichains in the Bruhat Order for the Classes $$\mathcal {A}(n,k)$$

Abstract

Let \(\varvec{\mathcal {A}(n,k)}\) represent the collection of all \(\varvec{n\times n}\) zero-and-one matrices, with the sum of all rows and columns equalling \(\varvec{k}\). This set can be ordered by an extension of the classical Bruhat order for permutations, seen as permutation matrices. The Bruhat order on \(\varvec{\mathcal {A}(n,k)}\) differs from the Bruhat order on permutations matrices not being, in general, graded, which results in some intriguing issues. In this paper, we focus on the maximum length of antichains in \(\varvec{\mathcal {A}(n,k)}\) with the Bruhat order. The crucial fact that allows us to obtain our main results is that two distinct matrices in \(\varvec{\mathcal {A}(n,k)}\) with an identical number of inversions cannot be compared using the Bruhat order. We construct sets of matrices in \(\varvec{\mathcal {A}(n,k)}\) so that each set consists of matrices with the same number of inversions. These sets are hence antichains in \(\varvec{\mathcal {A}(n,k)}\). We use these sets to deduce lower bounds for the maximum length of antichains in these partially ordered sets.