Some Identities On Partitions And Compositions With Restrictions On Consecutive Parts

The main results in this thesis are some identities concerned with partitions and compositions with restrictions on consecutive parts, including two Rogers-Ramanujan type identities for overpartitions, combinatorial proofs of two Rogers-Ramanujan type identities due to Andrews, an identity revealing a deep con-nection between anti-lecture hall compositions and overpartitions, a generating function formula for lecture hall partition with the first part bounded.Euler’s partition theorem, which states that the number of partitions of n with distinct parts equals the number of partitions of n with odd parts, is considered to be the first partition identity and foreshadows a number of partition identities.Since Euler’s partition theorem involves partitions in which the differences between consecutive parts is at least1, one family of partition theorems after Euler’s deal with partitions with restrictions on the differences between con-secutive parts or with restrictions on the occurrences of consecutive integers. Rogers-Ramanujan identity which concerns with2-distinct partitions is one of the most celebrated identities in this family. Many other mathematicians like Schur, Gordon, Gollnitz, Bailey, Slater, Andrews make contributions in finding new Rogers-Ramanujan type identities. Baxter, Andrews, Forrester introduce Rogers-Ramanujan type identities into physics to show that they are related to the hard hexagon model in statistical mechanics.Chapter2of this thesis is devoted to some Rogers-Ramanujan type identi-ties. First we discover two new Rogers-Ramanujan type identities for overparti-tions and prove them by employing a transformation formula due to Andrews. Furthermore, a combinatorial interpretation of one of these two identities by considering the successive Durfee dissection of overpartition is provided. We also find involutions for proving two Rogers-Ramanujan-Gordon type identities ob-tained by Andrews. The first is a Rogers-Ramanujan-Gordon type identity on the generating functions for partitions with part difference and parity restric-tions. The second contains both Euler’s identity and Rogers-Ramaujan-Gordon identity as direct consequences.Another family of identities after Euler’s deal with partitions or compositions with some restrictions on the ratio of consecutive parts instead of differences of consecutive parts. Lecture Hall Partition Theorem due to Bousquet-Melou and Eriksson is one of the most elegant theorems in this family. Corteel and Sav-age defined the anti-lecture hall composition, which can be seen as the twist structure of lecture hall partition, and obtained Anti-lecture Hall Composition Theorem. Some identities concerned with other sets of partitions or composi-tions constrained by the ratio of consecutive parts, such as truncated lecture hall partitions,(k,l)-lecture hall partitions, were also constructed.In Chapter3of this thesis, we deal with partitions and compositions con-strained by the ratio of consecutive parts. First we look deeper into the con-nection between anti-lecture hall compositions and overpartitions to show that the number of anti-lecture hall compositions of n with the first entry not ex-ceeding k-2equals the number of overpartitions of n with non-overlined parts not congruent to0,±1modulo k. This identity can be considered as a finite version of Anti-lecture Hall Theorem due to Corteel and Savage. To prove this result, we use some combinatorial method to calculate the generating function of anti-lecture compositions with the first entry not exceeding k-2depanding on the parity of k and employ two Rogers-Ramanujan type identities for overpar-titions given in Chapter2. When k is odd, we give another proof by using the bijections of Corteel and Savage for anti-lecture hall theorem and the generalized Rogers-Ramanujan identity due to Andrews. We also consider the relationship between lecture hall partitions and anti-lecture hall compositions to give a gener-ating function formula for the lecture hall partitions with the first part bounded. At the end of Chapter3, an identity on compositions with restrictions on ratioof consecutive parts and an extra restriction is derived.