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.