Partial vexillarity and bigrassmannian permutations

We define two closely related notions of degree for permutation patterns of type 2143. These give rise to classes of “m-vexillary elements” in the symmetric group. Using partitions, the Ehresmann–Bruhat partial order, and sets constructed from permutation inversions, we characterize the m-vexillary elements. We relate the maximal bigrassmannian permutations in the (Ehresmann–Bruhat) order ideal generated by any given m-vexillary element w to the maximal rectangles contained in the shape of w.