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{\bf William Y. C. Chen, Toufik Mansour and Sherry H. F. Yan}
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{\bf Matchings Avoiding Partial Patterns}
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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 Schr\"oder
paths without peaks at level one, which are counted by the
super-Catalan numbers or the little Schr\"{o}der 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 $12312$.
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