The philosopher Plato argued that there were two realms of existence: one consisting of perfected constructions of unseen but significant objects called Ideals and another consisting of the imperfect experiences of those objects. Within the realm of Ideals, the argument goes, exist numbers. While we all have an understanding of a number, like “eight,” none of us have actually experienced Plato's true interpretation of “eight.” We have only experienced various representations and agreements about what “eight" means. This leads us to the possibility of manipulating various representations of numbers for various effects. One classic representation of a number involves expressing an integer n as a sum of positive integers called a partition. This too, however, can be manipulated. To differentiate between distinct partitions of n, diagrams called Young tableaux are often used as representations. We will investigate unique properties of these representations to see that a special class of symmetric polynomial is housed within lattices of Young tableaux. Finally, we will consider unknown polynomials possessing multiple symmetries as a result of restricted and altered Young lattices.
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