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Partitions with short sequences and mock theta functions
1Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. andrews@math.psu.edu
Summary
This study explores integer partitions, specifically those without consecutive integers. These partitions have surprising applications in threshold growth models and connections to Ramanujan
Area of Science:
- Number Theory
- Combinatorics
- Mathematical Physics
Background:
- Integer partitions are fundamental in combinatorics.
- MacMahon's work initiated the study of partitions with restrictions on consecutive parts.
- Recent research links these partitions to threshold growth models.
Purpose of the Study:
- To intrinsically develop the theory of integer partitions without consecutive parts.
- To deepen the understanding of the applications of such partitions.
- To reveal the connection between these partitions and Ramanujan's mock theta functions.
Main Methods:
- Combinatorial analysis of partition properties.
- Exploration of connections to threshold growth models.
- Investigation of relationships with mock theta functions.
Main Results:
- Characterization of partitions with no sequences of consecutive integers of length k.
- Demonstration of the relevance of these partitions in threshold growth models.
- Identification of a surprising role for Ramanujan's mock theta functions.
Conclusions:
- The study of integer partitions with restricted consecutive parts offers a rich area for theoretical development.
- These partitions have unexpected applications in diverse fields like growth models.
- Ramanujan's mock theta functions play a significant, previously unrecognized role in this domain.
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