Arithmetic Properties For Restricted Partitions And The Study On Catalan Numbers

The theory of integer partitions is a subject of enduring interest from the era of the mathematician Euler in the 18th century.A major research area in its own right,it has found numerous applications,and celebrated results such as Ramanujan's congruences make it a topic filled with the true romance of mathematics.Catalan numbers are probably the most ubiquitous sequence of numbers in mathematics.There are some properties and applications in combinatorics,number theory,algebra,analysis,geometry,topology,and other areas.There are so many generalizations for classical Catalan numbers.The main objective of this thesis are two kinds of restricted partitions,namely,2-color partition triples as well as k-colored partitions,and a kind of?q,t?-Catalan numbers.We establish a number of Ramanujan-type congruences for these restricted partition functions.We also obtain some inequalities for k-colored partition functions.Moreover,we give several new combinatorial interpretations for this?q,t?-Catalan numbers along with corresponding gamma expansions using pattern avoiding permutations.The thesis is divided into seven chapters as follows:In Chapter 1,we introduce the mathematical background and the related progress of integer partitions as well as Catalan numbers.Moreover,we will briefly describe the main content of the present thesis.In Chapter 2,we establish several infinite families of congreunces modulo powers of 3 for p_3,1?n?,p_3,3?n?,and p_3,9?n?.Moreover,we also obtain two infinite families of congruences modulo 5 for p_5,1?n?and three infinite families of congruences for p_25,1?n?.Here_kp,₃?n?denote the number of 2-color partition triples of n where one of the colors appears only in parts that are multiples of k.In Chapter 3,we establish some infinite families of congruences modulo 25 for k-colored partition functionsp_-k?n?via elementary method.Furthermore,we prove some infinite families of Ramanujan-type congruences modulo powers of 5 forp_-k?n?with k?28?2,6,and 7.In Chapter 4,we obtain some inequalities for k-colored partition functions p_-k?n?for all integers k?2,which was motivated by a partition inequality of Bessenrodt and Ono involving ordinary partition function.In Chapter 5,we introduce a generalized crank?k-crank?for k-colored partitions.Following the work of Andrews-Lewis and Ji-Zhao,we derive two results for this newly defined k-crank.Namely,we first obtain some inequalities between the k-crank countsM_k?7?r,m,n?8?for m?28?2,3,and 4,then we prove the alternating sign property of symmetrized even k-crank moments weighted by the parity for k?28?2 and3.We also propose a unimodal conjecture involvingM_k?7?m,n?8?.In Chapter 6,we first prove several new combinatorial interpretations of a kind of?q,t?-Catalan numbers along with their corresponding?-expansions using pattern avoiding permutations.On the other hand,we give a complete characterization of certain?-1?-phenomenon for each subset of permutations avoiding a single pattern of length three,and further discuss their q-analogues utilizing the newly obtained q-?-expansions,as well as the continued fraction of a quint-variate generating function due to Shin and Zeng.In Chapter 7,we summarize the work in this paper and put forward the content of the future investigation.