counting patterns in a word Do algorithms exist to find all the patterns in a word? 
I would like to count all the 3-step increasing sequences (123 i.e. 123, 234 & 345) in some word in the alphabet {1,2,3,4,5} 
such as "51234321343221343234".  I don't care about spacing, so the subword 234 is good, but 23****4 is also good.
What if I ask these subwords be disjoint? I would expect a Robinson-Schensted-like algorithm except the increasing subsequences are of a fixed length. 
Do asymptotics exist in either case? 

How about patterns which are not necessarily increasing like 121 (i.e. 121,232,343,454,3**4*3, etc.)?

e.g. 51234321343221343234 has 343,121, 232,343,343 with 52124 left over.

$x = (x_1,x_2,\dots,x_3)$ and $y = (y_1,y_2,\dots,y_3)$ are "equivalent up to translation" if there is a single $k$ with $y_i = x_i + k$.  A "pattern" an equivalence class of words equivalent up to translation.  An "occurrence" of a pattern is an ordered subset of letters in a word equivalent to the given patterns. The bold numbers in 512 3 43213 4 32213432 3 4 are an occurrence the pattern 121.
 A: If you were asking for 3-step increasing sequences in permutations - and not words - then you are actually talking about so-called bivincular patterns. These were introduced in 2008 by Bousquet-Melou et al (http://arxiv.org/abs/0806.0666). Since you are only requiring that your sequences be consecutive in values - and don't place any restrictions on locations - then you could apply the inverse map (in terms of the permutation matrix it's reflection in a diagonal) to both the pattern and the permutation and you would turn your value-restrictions into location-restrictions. This would mean your patterns now become so-called vincular patterns. These were introduced in 2000 by Babson & Steingrimsson (http://combinatorics.cis.strath.ac.uk/download/BaSt00__Generalized_Permutation.pdf) under the name 'dashed patterns'.
There have been some algorithms written to count the number of permutations avoiding both types of patterns. See for example Nakamura (http://arxiv.org/abs/1102.2480).
Now, your question was about words, and not about permutations. I have not done any work in that area but there was a book that came out recently about patterns in words and permutations, see Sergey (http://www.springer.com/computer/theoretical+computer+science/book/978-3-642-17332-5)
