On q-series Identities Arising from Lecture Hall Partitions
نویسندگان
چکیده
In this paper, we highlight two q-series identities arising from the “five guidelines” approach to enumerating lecture hall partitions and give direct, qseries proofs. This requires two new finite corollaries of a q-analog of Gauss’s second theorem. In fact, the method reveals stronger results about lecture hall partitions and anti-lecture hall compositions that are only partially explained combinatorially.
منابع مشابه
Lecture hall sequences, q-series, and asymmetric partition identities
We use generalized lecture hall partitions to discover a new pair of q-series identities. These identities are unusual in that they involve partitions into parts from asymmetric residue classes, much like the little Göllnitz partition theorems. We derive a two-parameter generalization of our identities that, surprisingly, gives new analytic counterparts of the little Göllnitz theorems. Finally,...
متن کاملLecture hall theorems, q-series and truncated objects
We show here that the refined theorems for both lecture hall partitions and anti-lecture hall compositions can be obtained as straightforward consequences of two q-Chu Vandermonde identities, once an appropriate recurrence is derived. We use this approach to get new lecture hall-type theorems for truncated objects. The truncated lecture hall partitions are sequences (λ1, . . . , λk) such that λ...
متن کاملFive Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems
Five simple guidelines are proposed to compute the generating function for the nonnegative integer solutions of a system of linear inequalities. In contrast to other approaches, the emphasis is on deriving recurrences. We show how to use the guidelines strategically to solve some nontrivial enumeration problems in the theory of partitions and compositions. This includes a strikingly different a...
متن کاملOverpartition Theorems of the Rogers-ramanujan Type
We give one-parameter overpartition-theoretic analogues of two classical families of partition identities: Andrews’ combinatorial generalization of the Gollnitz-Gordon identities and a theorem of Andrews and Santos on partitions with attached odd parts. We also discuss geometric counterparts arising from multiple q-series identities. These involve representations of overpartitions in terms of g...
متن کاملCombinatorial Proofs of q-Series Identities
We provide combinatorial proofs of six of the ten q-series identities listed in [3, Theorem 3]. Andrews, Jiménez-Urroz and Ono prove these identities using formal manipulation of identities arising in the theory of basic hypergeometric series. Our proofs are purely combinatorial, based on interpreting both sides of the identities as generating functions for certain partitions. One of these iden...
متن کامل