  A permutation is called Grassmannian if it has at most one descent. The study
of pattern avoidance in such permutations was initiated by Gil and Tomasko in
2021. We continue this work by studying Grassmannian permutations that avoid an
increasing pattern. In particular, we count the Grassmannian permutations of
size $m$ avoiding the identity permutation of size $k$, thus solving a
conjecture made by Weiner. We also refine our counts to special classes such as
odd Grassmannian permutations and Grassmannian involutions. We prove most of
our results by relating Grassmannian permutations to Dyck paths and binary
words.
