Matchings Avoiding Partial Patterns

William Y. C. Chen1, Toufik Mansour, and Sherry H. F. Yan

  Abstract:  We show that matchings avoiding certain partial patterns 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 nd a bijection between Schroder paths without peaks at level one and matchings avoiding both patterns 12312 and 121323. Such objects are counted by the super-Catalan numbers or the little Schroder numbers. A re nement of the super-Catalan numbers is obtained by xing the number of crossings in the matchings. In the sense of Wilf-equivalence, we nd that the patterns 12132, 12123, 12321, 12231, 12213 are equivalent to 12312.

  AMS Classification:  05A05, 05C30.

  Keywords:  Generating function, generating tree, matching, ternary tree, super-Catalan number, oscillating tableau.

  Download:   PDF