The Research Of Several Partition Identities And Inequalities

Since Euler gave the famous partition theorem,integer partition has attracted people to study.As a hot field in combinatorial mathematics,partition identities and inequalities have been concerned by many scholars.In this thesis,we mainly study partition functions based several statistics,establish some partition identities and some partition inequalities,some of which we give combinatorial proofs.The main research contents are as follows:First,Andrews et al.proposed the k-measure statistic and obtained the conclusion that the 2-measure and Durfee square side lengths are same distribution.We introduce partition length and minimum part,and obtain a refinement of them in this theorem.Second,for the minimal excludant statistics,we consider the number of parts larger than the minimal excludant and the number of parts smaller than the minimal excludant in the partition,and obtain two new inequalities.At the same time,we are also try to conduct research on the r-gap proposed by Ballantine and Merca.Thirdly,inspired by the work of Kim et al.,based on the parity of some statistics,we establish some inequalities on bipartitions.Fourth,we consider the 2-measure in the distinct partition,and obtain the generalizations of Bressoud et al.conclusion on the partitions into 2-distinct parts by using the two methods of analysis and combination.Fifthly,we construct a combinatorial technique to prove some Ramanujan’s identities.