  We give an explicit formula for the degree of the Grothendieck polynomial of
a Grassmannian permutation and a closely related formula for the
Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a
Grassmannian permutation. We then provide a counterexample to a conjecture of
Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard
open patches of particular Grassmannian Schubert varieties and show that our
work gives rise to an alternate explicit formula in these cases. We end with a
new conjecture on the regularities of standard open patches of arbitrary
Grassmannian Schubert varieties.

Combinatorics;Commutative Algebra