  In this paper, we extend to a non-consecutive case, the study of the pattern
matching condition on the wreath product of the cyclic group and the symmetric
group initiated by the authors in a previous paper. The main focus of our paper
is (colored) patterns of length 2, although a number of enumerative results for
longer patterns are also presented. A new non-trivial bijective interpretation
for the Catalan numbers is found, in terms of simultaneously bi-avoiding two
patterns in a wreath product.
