Discrete Entropy Power Inequalities via Sperner Theory

A ranked poset is called strongly Sperner if the size of k-family cannot exceed the sum of k-largest Whitney numbers. In the sense of a function ordering, a function f is (weakly) majorized by g if the the sum of k-largest values in f cannot exceed the sum of k-largest values in g. Two definitions arise from different contexts, but they have strong similarities in their own meanings. Furthermore, the product of weighted posets has another similarity with the convolution of functions. Elements in the product of ranked and weighted posets capture structures of the building blocks in the convolution. Combining all together, we are able to derive various types of entropy inequalities and discuss their applications.