Distributions of mesh patterns of short lengths on separable permutations

This paper contributes to the long line of research on the distribution of mesh patterns in permutations. We extend these studies to separable permutations and carry out a comprehensive analysis of mesh patterns of length at most~2. For mesh patterns of length~1, we determine the distributions for all six equivalence classes. In addition, we obtain the joint distribution for the patterns in the class containing the well-known permutation statistic known as the strict fixed point. Furthermore, computer experiments suggest that at most 124 pairs of mesh patterns of type 12 and 21 with identical shading are equidistributed. Using symmetry operations, we partition these pairs into 38 equivalence classes with respect to distribution. By explicitly determining 24 of these distributions (in many cases, in fact, joint equidistributions of the respective pairs), we reduce the number of equivalence classes to~31, which turns out to be the true number of equivalence classes. Enumerating 14 of these classes is left as an open problem.