Partition Identities Related To The Minimal Excludant

The theory of integer partitions is a very important subject in Combinatorics,which is also an integral part in Number Theory.What people concern most about the partition theory is to establish the equality between two different kinds of partitions,which we call partition identities.In 2019,Professor Andrews defined a new statistic,called the minimal excludant,to keep track of the missing smallest part in a partition.In this thesis,we focus on investigating further some classical partition identities with the help of the minimal excludant.The structure is organized as follows.Firstly,we obtain a refinement of the partition theorem related to the minimal exclu-dant by considering the length of partitions,which was found by Andrews.Secondly,we study the minimal odd excludant in partitions and get some new results about the known partition identities.For ordinary partitions,we present an identity about the statistics "minimal odd excludant" and "length".By relating these to other statistics,we propose two new refinements of this result.Moreover,we explore the minimal odd excludant for partitions into odd parts,and propose one partition identity,which can be regarded as an analogue to Andrews’ partition theorem.Thirdly,we refine the celebrated Euler’s partition theorem by researching the residue classes of the minimal odd excludant modulo 4.Meanwhile,we find a similar identity for the partitions into distinct parts with the smallest part odd,which is a new refinement of Fine’s partition theorem.Both the analytic and combinatorial proofs of this result are provided.In addition,we construct a bivariate form of the mock theta function φ(-q),and a combinatorial interpretation of the symmetry about the famous statistic "crank".Finally,the main work of this thesis is summarized,and some possible further re-search work is discussed.