Distributions of mesh patterns of short lengths

A systematic study of avoidance of mesh patterns of length 2 was conducted in [I. Hilmarsson et al., Wilf-classification of mesh patterns of short length, Electr. J. Combin. 22(4) (2015), \#P4.13.], where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove 2 equidistribution results (out of 7 equidistribution results we prove in total), and conjecture 7 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic "strict fixed point", which plays a key role in many of our enumerative results.
This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to obtaining functional relations for generating functions, and finding recurrence relations and bijections.