Inversion Sequences Avoiding Pairs Of Patterns

The enumeration of inversion sequences avoiding a single pattern was initiated by Corteel,Martinez,Savage and Weselcouch,and Mansour and Shattuck independently.Their work has sparked various investigations of generalized patterns in inversion sequences,including patterns of relation triples by Martinez and Savage,consecutive patterns by Auli and Elizalde,and vincular patterns by Lin and Yan.In this thesis,we carried out the systematic study of inversion sequences avoiding two patterns of length 3 and we also obtain the classification of the Wilf equivalence.Our Wilf equivalence classifications establish further connections to the OEIS sequences and some classical combinatorial objects,such as restricted permutations,weighted ordered trees and set partitions.The first chapter mainly introduces the basic concept,research background and main content of this thesis.In the second chapter,by analyzing or constructing simple bijection we prove that the enumeration of the inversion sequences of the 11 types of avoiding pairs of patterns turn out to be the classical combinatorial numbers,such as Fibonacci numbers,Catalan numbers and Schroder numbers.One of Martinez and Savage's conjectures regarding the enumeration of(021,120)-avoiding inversion sequences is solved in chapter 3.In addition,we obtain the number of the inversion sequences avoiding pairs of(110,102)and(102,120)by the kernel method.In the final chapter,we enumerate 4 types of inversion sequences avoiding pairs of patterns by constructing a bijection or establishing identical recursion and obtain the relations with Boolean permutation,set partitions and indecomposable set partitions,etc.