On this project, we explored the behavior of the new notion of global pattern containment introduced here.

We consider a signed permutation w as a bijection on the set 

{-n,-n+1...-2,-1,1,2...n} such that w(-i)=-w(i). Since signed permutations are odd, the are uniquely defined by their positively indexed values. For example, the signed permutation w={-2,1,3,-4} is plotted to the left. By isolating a size k subsequence in a signed permutation and relabing the values with {1,2...,k} while preserving order, we obtain a global pattern contained in w. For example, since {-2,1,3,-4} contains a subsequence, (-3,2,1,3), order isomprphic to (1,3,2,4), we say {-2,1,3,-4} globally contains (1,3,2,4).

Our research studied the sets of signed permutations which don't contain particular global patterns. We considered subtle set bijections, containments, redundancies, and enumerations. Not only does a rich, vibrant, and nuanced theory exist for global pattern containment, the notion well classifies a wide variety of important combinatorial objects. For example, the signed permutations which contain no 2143 global pattern are the type B vexillary permutations, and the signed permutations containing no 321 or 3412 global patterns are the Type B Boolean permutations.