  The notion of a mesh pattern was introduced recently, but it has already
proved to be a useful tool for description purposes related to sets of
permutations. In this paper we study eight mesh patterns of small lengths. In
particular, we link avoidance of one of the patterns to the harmonic numbers,
while for three other patterns we show their distributions on 132-avoiding
permutations are given by the Catalan triangle. Also, we show that two specific
mesh patterns are Wilf-equivalent. As a byproduct of our studies, we define a
new set of sequences counted by the Catalan numbers and provide a relation on
the Catalan triangle that seems to be new.
