Partition Identities Through the Weighted Words Approach

In mathematics and computer algebra, factorization (of integers, polynomials, and series) is one of the fundamental components. It gives a connection between additively written information and multiplicatively written information. To be able to find different representations of an object is mathematically valuable and computationally effective. For example, people and computers can do addition much faster than multiplication, on the other hand, knowing the factors of an object is analogous to knowing the exact atoms that makes a molecule. In this project, we focus on integer partitions (additive representations of whole numbers) and search for profound mathematical connections among them. Partition identities is a research area that lies in the intersection of combinatorics, number theory, special functions, representation theory with connections to physics and computer science. In 1750s Euler noted the first known partition identity. Although partition identities have been of interest since then, it is still not clear when two sets of partitions can be shown to have the same number of elements. It is unclear under what conditions are needed to make a set of partitions with difference conditions equinumerous to a set of partitions with congruence conditions. Such identities are traditionally called Roger-Ramanujan type identities and they manifest themselves as a mathematical identity where an infinite series can be factored and that it is equal to an infinite product. In this project, we plan to start from a combinatorial technique Krishnaswami Alladi and Basil Gordon have introduced in late 1990s, namely the method of weighted words, and systematically search a wide base of partitions to find new partition identities. The method of weighted words has been used by many leading researchers in the recent years but as a combinatorial technique its reach was limited by error-prone calculations. Recently, Jehanne Dousse revisited this technique and noted a new version that is suitable for algorithmic treatment. This allows us to carry all the calculations on computer algebra systems and to expand our horizons far beyond human reach. The principal investigator, Ali Uncu, together with Jakob Ablinger implemented the original version Dousses technique in a computer algebra system and showed that it can be used to prove some already known Rogers-Ramanujan type identities. This project aims to automate the experimental search and automatic proving of partition identities through the method of weighted words. Among new partition identities, this project plans to provide researchers around the globe high-quality free computer algebra implementations and it also plans to generalize domain where the weighted words can be used to larger classes of objects such as cylindric partitions and plane partitions. Jehanne Dousse, Christoph Koutschan, and Sylvie Corteel are our main project partners, whom we will be collaborating on our research topics.

This project allowed us to uncover many new relations between seemingly unrelated mathematical structures using symbolic computation tools. An integer partition, which is a tool to dissect and study numbers' additive properties. Integer partitions appear in every field of mathematics and high-energy physics, and doing mathematics with them generates a lot of computational challenges. Our project mainly focused on integer partition identities, where the number of elements from two different sets of integer partitions is equal. There are still many open problems in this field. For example, the Ising model, which studies the statistics of magnetism of atoms (in physics), probability distribution on discrete sets (in probability), opinion dynamics (in sociology), etc. It is known that the integer partition identities of certain kinds make up the character formulas of Ising models, but there are only a handful of character formulas out of infinitely many models that we know and can explain at the moment. These identities are rare and incredibly difficult to discover. We built new experimental and symbolic computation tools that would allow researchers worldwide to search for and prove new partition identities. We have built many new algorithms and disseminated their capabilities by proving new identities with them. The collaborative work on the cylindric partitions and our new modulo 8 identities led to the discovery of a family of partition identities. That research has been gaining attention and momentum ever since. The principal investigator was funded by the Austrian Science Fund's SFB F50-11 project while carrying out that work. That initial proof would not have been possible without our computer implementations. The project P34501-N builds on those tools that have been developed and increases our mathematical reach even more. Among these is the proof of the modulo 11 and 13 cylindric partition identities, which were discovered following the modulo 8 identities, but couldn't have been proven without the tools that were built within this project. The project allowed the investigator to contribute to other mathematical questions, too. In a series of works on simplifying logical systems with constraints, the principal investigator and collaborators applied and refined existing algebraic geometry techniques to new problems. With collaborators from the Symbolic Computation group of the Johann Radon Institute of the Austrian Academy of Sciences (RICAM), the investigator applied these techniques to understand new properties of the number of integer partitions in a restricted box and answered some open problems. The investigator also collaborated with the Computational Methods group member of RICAM to simplify some optimization calculations for the industry. The project generated 20 scientific articles, most of which are now published in high-standing international refereed journals and conference proceedings as of the end of the project.