Mathematics, Models, and Magz, Part II: Investigating Patterns in Pascal's Simplex

摘要

This paper began when one of the authors, JP, gave a colloquium talk at Santa Clara University on the content of "Mathematics, Models, and Magz, Part I" describing results later reported in this Magazine [5]. She described the "Star of David" theorems, which involve two-dimensional arrays of binomial coefficients, and their three-dimensional analogues involving trinomial coefficients. The other author, VG, then a sophomore at Santa Clara, was in the audience and became interested in seeing what would happen in four dimensions. The authors applied for, and received, a grant from Santa Clara University's Student Research Initiative Program that allowed them to spend the summer of 2009 studying extensions of the results in two and three dimensions.