Partition Statistics And Beck Type Identities

The theory of integer partitions is a fascinating branch of combinatorics,which dates back to the celebrated partition theorem proposed by Euler in 1748.In 2017,George Beck posed two conjectures concerning the distribution of the number of parts and the number of distinct parts on the partitions involved in Euelr's partition theorem.We say that an identity is a Beck type identitiy if its form is similar to that in Beck's conjectures.In this thesis,we mainly study the Beck type identities on partitions from the following aspects:Firstly,we present a new more general form of Beck's conjectures corresponding to Franklin's theorem,which includes the original conjectures and some known generalizations as its special cases.Meanwhile,we give new combinatorial interpretations of Euler's partition theorem and Glaisher's theorem by introducing two pair of partition sets with equal size.The results are proved both analytically and combinatorially.Secondly,we establish new and simple bijections to prove the composition analogues of Euler's,Glaisher's,and Franklin's partition theorem respectively.Moreover,with the help of the statistics the number of odd parts and the number of parts greater than one,we obtain two Beck type identities on compositions and their generalizations.Relying on our bijections,we explain these identities combinatorially.Thirdly,we generalize Ramanujan's partition theorem,which is deduced from two fifth order mock theta functions,in a combinatorial manner.Furthermore,based on our general partition identity,we present two Beck type identities according to the length and the number of distinct parts.Apart from the traditional q-series proofs,we also prove the main results bijectively.Fourthly,we give more direct bijections to show the new partition identities associated with the total number of even parts in distinct partitions,which are derived by Andrews and Merca recently.The partitions enumerated by the number of even parts in distinct partitions are revealed under our bijections.Finally,we review some recent work related to Beck type identities,and discuss certain future topics which are worth further investigations.