Baxter d-Permutations and Other Pattern-Avoiding Classes

A permutation of size n can be identified with its diagram in which there is exactly one point in each row and column in the grid [n](2). In this paper we consider multidimensional permutations (or d-permutations), which are identified with their di-agrams on the grid [n](d) in which there is exactly one point per hyperplane x(i) = j for i is an element of [d] and j is an element of [n]. We first exhaustively investigate all small pattern-avoiding classes for d = 3. We provide several bijections to enumerate some of these classes and we propose conjectures for others. We then give a generalization of the well-studied Baxter permutations to higher dimensions. In addition, we provide a vincular pattern -avoidance characterization of Baxter d-permutations.