Bijective Enumerations Of Tiered Trees And Core Partitions

Enumerative combinatorics is an important research direction in combinatorics,which is mainly corncerned with discrete structures.Exhibiting some mathematical problems elegantly and intuitively by virtue of combinatorial methods is one of the hottest problems in enumerative combinatorics.It has attracted extensive attention from researchers.The essence of the combi-natorial methods lies in constructing the combinatorial objects and establishing the appropriate combinatorial transformations.Tiered trees were introduced by Dugan-Glennon-Gunnells-Steingrimsson as a general-ization of intransitive trees that were introduced by Postnikov.Tiered trees arise naturally in counting the absolutely indecomposable representations of certain quivers,and enumerating torus orbits on certain homogeneous varieties over finite fields.Integer partition is also an important research object in enumerative combinatorics.It has important applications in group theory,probability theory,mathematical statistics and particle physics.Core partition was raised in the study of modular theory of the representation theory of the symmetry group.It is closely related to various kinds of structures including rational Dyck paths,posets,parking functions,hyperplane arrangements and Coxeter group.In this thesis,on the one hand,we study the bijective enumeration of tiered trees.By em-ploying generating function arguments and geometric results,Dugan et al.derived an elegant formula concerning the enumeration of tiered trees,which is a generalization of Postnikov's formula for intransitive trees.We provide a bijective proof of this formula by establishing a bijection between tiered trees and certain rooted labeled trees.As an application,our bijection also enables us to derive a refinement of the enumeration of tiered trees with respect to level of the node 1.On the other hand,we deal with the enumeration of self-conjugate(s,s+d,s+2d)-core partition.Ford,Mai and Sze established a bijection between self-conjugate(s,t)-core partitions and lattice paths in the[s/2]×[t/2]rectangle consisting of north and east steps,thereby showing that the number of such partitions is given by((?))for relatively prime integers s and t.In this thesis,we explore self-conjugate(s,s+d,s+2d)-core partitions in the spirit of the work of Ford,Mai and Sze.We provide a lattice path interpretation of self-conjugate(s,s+d,s+2d)-core partitions in terms of free Motzkin paths and obtain the enumeration of such core partitions.