Geodesic
The Complexity of Pattern Matching for $321$-Avoiding and Skew-Merged Permutations
Discrete mathematics & theoretical computer science, Permutation Patterns 2015, Tome 18 (2015-2016) no. 2.

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The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $\pi$ and $\tau$ are $321$-avoiding; the second is applicable if $\pi$ and $\tau$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $\pi$ and $n$ the length of $\tau$. 
@article{DMTCS_2016_18_2_a9,
     author = {Albert, Michael H. and Lackner, Marie-Louise and Lackner, Martin and Vatter, Vincent},
     title = {The {Complexity} of {Pattern} {Matching} for $321${-Avoiding} and {Skew-Merged} {Permutations}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2015-2016},
     doi = {10.46298/dmtcs.1308},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.1308/}
}
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Albert, Michael H.; Lackner, Marie-Louise; Lackner, Martin; Vatter, Vincent. The Complexity of Pattern Matching for $321$-Avoiding and Skew-Merged Permutations. Discrete mathematics & theoretical computer science, Permutation Patterns 2015, Tome 18 (2015-2016) no. 2. doi : 10.46298/dmtcs.1308. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.1308/

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