The Congruence Properties Of Two Types Of Partition Functions With Constraints

In 1919,three famous congruences for partition functions were discovered by Ramanujan.Since then,congruence properties for restricted partition function is one of the most important topic in the theory of integer partitions and a number of congruences for restricted partition function were proved.In this thesis,we investigate congruence properties of two restricted partition functions:the number of tagged parts over the partitions with designated summands and partitions wherein each part appears an odd number of times.In 2002,a new class of partitions,partitions with designated summands,was introduced by Andrews,Lewis and Lovejoy.Recently,Lin introduced the partition function PDt(n),which counts the total number of tagged parts over all the partitions of n with designated summands.Lin also proved some congruences modulo 3 and 9 for PDt(n)and posed a conjecture on congruences modulo 9 for PDt(27n+6)and PDt(27n+21).In this thesis,we confirm and generalize Lin’s conjecture based on the formulas for r2(n)and r6(n),where rk(n)is the number of representations of n as sums of k squares.Let f(n)be the number of partitions of n in which each part appears an odd number of times.In a recent work,Hirschhorn and Sellers proved a characterization of f(2n)modulo 2,which implies infinitely many Ramanujan-like congruences modulo 2 satisfied by the function f(n).In this paper,we discovered some new parity results for f(2n+1).In particular,we prove some nonlinear congruences modulo 2 for f(n).For example,we prove that for n≥0.f(2340n2+780n+65)≡(2340n2+3900n+1625)≡0(mod 2).