Study On Congruence Properties Of Restricted Partition Functions

Integer partition is an important research object in combinatorics.The study on integer partition date back to Euler,and since then many mathematicians such as Jacobi?Gauss?Sylvester?MacMahon have done a lot of research about it.Ramanujan,an Indian mathematician,pioneered the study of congruence properties of partition functions and proposed some related conjectures.In recent years,mathematicians led by Andrews?Bendt and Ono have greatly enriched the research content of this field,making it gradually become a research focus at present.It is worth mentioning that the introduction of modular forms has made a great breakthrough in the study of congruence properties.At present,the congruence properties of restricted partition functions are widely concerned,and its research tools involve combinatorial methods?q-series and modular forms.In this thesis,the related congruence properties of several restricted partition functions are studied by using the theory of q-series.Some identities of q-series and related dissection formulas are the key to studying the congruences of partition functions.In the first chapter,we introduce the research background,including the history of research and current situation.In the second chapter,we briefly introduce the basic conception of integer partition?some classical formulas in the theory of q-series and several dissection formulas.In the third chapter,we firstly consider two kinds of restricted partition functions introduced by Choi and Kim.We obtain the 2-dissection formula of partition function b(n)and prove some congruences modulo 2.For 3-core cubic partition function d(n),we establish two congruences modulo 2 and 9 by using dissection formulas given by Chan.After that,we investigate two partition functions related to mock theta functions and give new proofs for several known congruences.In the forth chapter,we focus on the arithmetic properties of regular bipartition function Bk,l(n).For(k,l)?{(4,4),(3,5),(3,7),(5,7)},We study the congruence properties of Bl,l(n)respectively and obtain some congruences modulo 2,3,4 and 5.In particular,we confirm an open problem of B3,7(n)modulo 3,which was proposed by Dou.In addition,for a class of partition functions dk,l(n)and(k,l)?{(2,5),(2,7),(2,11),(3,4),(4,5),(5,8)},we also obtain some congruences modulo 2 and 4.In the fifth chapter,we study the arithmetic properties of several new classes of restricted partition fucntions.More precise,we first establish several infinite families of congruences modulo 5 and 7 for the partition fucntion lk(n).Then,for the partition function bk(n),we obtain several infinite families of congruences modulo 2 and 5.We also obtain several congruences modulo 3,5,7 and 11 for the partition function c(n).Finally,we present several congruences modulo 2,5,6 and 7 for the partition function t(n).