Geodesic
Matchings avoiding partial patterns
The electronic journal of combinatorics, Tome 13 (2006)

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We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers. We give a characterization of $12312$-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns $12312$ and $121323$ and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of generating trees to show that the patterns 12132, 12123, 12321, 12231, 12213 are all equivalent to the pattern $12312$.
DOI : 10.37236/1138
Classification : 05A05, 05A15, 05C30
Mots-clés : oscillating tableaux, Schröder numbers, Schröder paths, super-Catalan numbers 
William Y. C. Chen; Toufik Mansour; Sherry H. F. Yan. Matchings avoiding partial patterns. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1138
@article{10_37236_1138,
     author = {William Y. C. Chen and Toufik Mansour and Sherry H. F. Yan},
     title = {Matchings avoiding partial patterns},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1138},
     zbl = {1112.05001},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1138/}
}
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