([I asked this ten days ago](https://math.stackexchange.com/questions/50051/stricter-permutation-patterns) on math.SE, but I received no reply, so I'm trying again here.)


A lot of work has been done on [patterns in permutations](http://en.wikipedia.org/wiki/Permutation_pattern), where a permutation is said to match a given pattern if it contains a subsequence of elements ordered according to the pattern (e.g., $\pi=(2\ 1\ 7\ 5\ 3\ 6\ 4)$ matches $4\ 1\ 3\ 2$ -- consider the subsequence $7\ 3\ 6\ 4$).


One could be more restrictive and require that the subsequence of elements to be tested against the pattern not only follows the same order, but that the gaps between the elements (or between some, but not all, pairs of elements) is the same. In the above example, for instance, $7\ 3\ 6\ 4$ would no longer be a match for $4\ 1\ 3\ 2$ (but $6\ 3\ 5\ 4$ would, had it been a subsequence of $\pi$).


Has there been work on such constrained patterns? If so, what are they called, and what would be good references to check?



Stricter permutation patterns
Yes, these things have been studied extensively in two ways:


1) As bivincular patterns, introduced in a 2010 paper by Bousquet-Mèlou, Claesson, Dukes and Kitaev.


2) If you require the occurrence in the permutation to be completely consecutive in values (like 6354) then the inverse of your pattern is called a consecutive pattern. These have been studied by Elizalde and many others.


