  In this paper, we investigate pattern avoidance of parity restricted (even or
odd) Grassmannian permutations for patterns of sizes 3 and 4. We use a
combination of direct counting and bijective techniques to provide recurrence
relations, closed formulas, and generating functions for their corresponding
enumerating sequences. In addition, we establish some connections to Dyck
paths, directed multigraphs, weak compositions, and certain integer partitions.

Combinatorics