On Self-conjugate Core Partitions

Integer partition is one of the most important research directions in combinatorics.It has been widely used in the representation theory,number theory and the theory of symmetric function.A particularly prominent theme is the study of core partitions.In recent years,core partitions become a research focus in combinatorics.It turns out that the study of core partitions is closely related to a variety of objects including rational Dyck paths and posets.Amdeberhan conjectured that the number of(s,s+1,s+2)-core partitions is equal to the number of Motzkin paths of length s,and posed the conjectures concerning the largest size and the average size of such core partitions.This conjecture was first confirmed by Yang,Zhong and Zhou.Amdeberhan and Leven further proved that the number of(s,s+1,...,s+k)-core partitions is equal to the number of(s,k)-Dyck paths.This paper is mainly concerned with self-conjugate(s,s+1,...,s+k)-core partitions and symmetric(s,k)-Dyck paths.we also evaluate the largest size of a self-conjugate(s,s+1,...,s+k)-core partition for given positive integers s and k.In Chapter 1,we mainly introduces the research background,preliminaries and our main results.In Chapter 2,we mainly provide a bijection between self-conjugate(s,s+1,...,s+k)-core partitions and symmetric(s,k)-Dyck paths.This generalizes the result of Amdeber-han and Leven to self-conjugate core partitions.In Chapter 3,we shall evaluate the largest size of a self-conjugate(s,s+1,...,s+k)-core partition for given positive integers s and k.