Pattern-avoiding even and odd Grassmannian permutations

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.