  We show that bigrassmannian permutations determine the socle of the cokernel
of an inclusion of Verma modules in type $A$. All such socular constituents
turn out to be indexed by Weyl group elements from the penultimate two-sided
cell. Combinatorially, the socular constituents in the cokernel of the
inclusion of a Verma module indexed by $w\in S_n$ into the dominant Verma
module are shown to be determined by the essential set of $w$ and their degrees
in the graded picture are shown to be computable in terms of the associated
rank function. As an application, we compute the first extension from a simple
module to a Verma module.
