On (S,t)-core Partitions

Integer partition theory is one of the most important research direction in combi-natorics,which has extensive application in representation theory,number theory and symmetric function.A particularly prominent theme is the study of core partitions.This research direction has received extensive attention from many famous combinatorialists.It turns out the study of core partitions is closely related to a variety of objects including Dyck paths,posets,rational simplex,Shi arrangement.A partition ? of a positive integer n is defined to be a sequence of nonnegative integers(?1,?2,…,?m)such that ?1??2?… +?m=n and ?1 ? ?2 · · · ? ?m.The empty partition is denoted by(?).We write ?=(?1,?2,...,?m)n and we say that n is the size of ?,denoted by ???.The Young diagram of ? is defined to be an up-and left-justified array of n boxes with ?i boxes in the i-th row.The hook of each box B in A consists of the box B itself and boxes directly to the right and directly below B.The hook length of B,denoted byh(B),is the number of boxes in the hook of B.For a partition A,the ?-set of ?,denoted by ?(?),is defined to the set of hook lengths of the boxes in the first column of ?.Notice that a partition A is uniquely determined by its ?-set.For a positive integer t,a partition is said to be a t-core partition,or simply a t-core,if it contains no box whose hook length is a multiple of t.Let s be a positive integer not equal to t,we say that A is an(s,t)-core partition if it is simultaneously an s-core and a t-core.In the last decades,(s,t)-core partitions have been extensively studied.In 2001,Anderson showed that the number of(s,t)-core partitions is given by the rational Catalan number 1/s+t(s+t s)when s and t are coprime to each other.The proof of Anderson's theorem is through characterizing the ?-sets of(s,t)-core partitions as order ideals of the poset Ps,t where(?)whose partial order is fixed by requiring x ?Ps,t to cover y ?Ps,t if x-y is either s or t.In 2009,Olsson and Stanton proved that the largest size of(s,t)-core partition is given by(?)when s and t are coprime to each other,which confirmed the conjecture posed by Aukerman,Kane and Sze.In 2014,Armestrong,Hanusa and Jones conjectured that the average size of(s,t)-core partition((s,t)-core self-conjugate partition)is given by(s+t+1)(s-1)(t-1)/24 if s and t are coprime to each other.Stanley and Zanello proved that the conjecture holds for(s,s + 1)-core partitions.Later,Aggarwal generalized the results of Stanley and Zanello,and showed that the conjecture holds for(k,mk +1)-core partition.Recently,people began to carry out research on core partition with restrictions.In 2009,Ford,Mai and Sze showed that the number of(s,t)-core self-conjugate partition is(?)Wwhen s and t axe coprime to each other.Chen,Huang and Wang proved that the number of(s,t)-core self-conjugate partition is given by(?)when s and t are coprime to each other,confirming the conjecture posed by Armestrong,Hanusa and Jones.Recently,Straub and Xiong independently showed that the number of(s,s + 1)-core partitions into distinct parts is given by the Fibonacci number Fs+1,which verifies a conjecture posed by Amdeberhan.Xiong also obtained the largest size and the average size of such partitions respectively,which completly settles Amdeberhan's conjecture concerning the enumeration of(s,s + 1)-core partitions into distinct parts.Analogous to the study of(s,t)-core partitions,people carried out the relevant research for multi-core partitions and have obtained a variety of results on the number,the largest size and the average size of such partitions.For example,Yang,Zhong and Zhou has derived the number,the largest size and the average size of(s,s + 1,s + 2)-core partitions.In this thesis,we are mainly concerned with the enumeration of(s,s + 2)-core parti-tions with distinct parts.We derive the number and the largest size of such partitions for the case when s is odd,confirming two conjectures posed by Straub.The thesis consists of four chapters.In Chapter 1,we mainly introduces the basic concepts,relevant research background,and our main results.In Chapter 2,we deal with the enumeration of(s,s + 2)-core partitions into distinct parts.We show that the number of such partitions equals 2s-1 when s is odd.In Chapter 3,we derive that the largest size of(s,s + 2)-core partitions into distinct parts is given by(?)when s is odd.In Chapter 4,we have some concluding remarks and pose some open problems.