## Description

The Ingleton Inequality, which is based on a vector space theorem and adapted to entropy functionals by Matu___, plays a role in describing the entropy region for four discrete random variables, depending on whether or not it is satisfied. The entropy regions that are defined by entropic vectors for n >_ 4 discrete random variables are not known, in general. For n = 4, recent work has been accomplished by Mao and Boston in characterizing this space of (2[superscript lower case n] --1 )-dimensional entropic vectors. They study how group-characterizable entropic vectors sometimes yield violations of the Ingleton Inequality/Ratio, thus expanding the understanding of this region. We discovered a decomposition of the Ingleton Ratio into what we dub the Mixing Ratio and the Inclusion-Exclusion Ratio, which resembles the inclusion-exclusion principle from set theory. We propose two conjectures based on the simpler Inclusion-Exclusion and Mixing Ratios, attempting to streamline the search for Ingleton Inequality-violators. Examples of these violators are reproduced in order to provide support for our conjectures. Characterizing the entropy region associated with n discrete random variables is related to determining the maximum communication rate of a network coding system that uses n discrete random variables. Characterizing this region for arbitrary n remains an open problem