Pattern Avoidance In Matchings And Involutions

This thesis concerns several enumerative results on matchings and involutions with restrictions. The thesis consists of three parts. The first part contributes to the enumeration problem of matchings (involutions without fixed points) avoiding partial patterns. The second part contributes to the problem of involutions with forbidden patterns. While the last part contributes to the combinatorial explanations of an identity on partial Motzkin paths and identities on noncrossing trees, symmetric ternary trees and noncrossing graphs.The first part consists of Chapter 2 and Chapter 3. In Chapter 2, we are interested in the enumeration of a class of matchings avoiding certain partial patterns. The following results are obtained.1. The number of 12312-avoiding matchings on [2n] with exactly m crossings is given by2. The number of 12312-avoiding matchings on [2n] is equal to 3-Catalan number C_n,3.3. Matchings avoiding both patterns 12312 and 121323 are in one-to-one correspondence with Schroder paths without any peak at level one. Such paths are counted by the super-Catalan numbers or the little Schroder numbers.4. The number of matchings on [2n] avoiding both patterns 12312 and 121323 with m crossings is equal towhich is a refinement of the super-Catalan numbers.5. By applying Stanley's bijection between oscillating tableaux and 12312-avoiding matchings, we derive a characterization of the oscillating tableaux for 12312-avoiding matchings.