Partition-Symmetrical Entropy Functions

Let N = (1,..., n). The entropy function h of a set of n discrete random variables (X_i : i ∈ N ) is a 2ⁿ-dimensional vector whose entries are h(A) △ H(X ), A c N, the (joint) entropies of the subsets of the set of n random variables with H(X_Ø) = 0 by convention. The set of all entropy functions for n discrete random variables, denoted by Γ_n*, is called the entropy region for n. Characterization of Γ_n* and its closure Γ_n* are well-known open problems in information theory. They are important not only because they play key roles in information theory problems but also they are related to other subjects in mathematics and physics. In this paper, we consider partitionsymmetrical entropy functions. Let p = (N₁,..., N_t) be a t-partition of N. An entropy function his called p-symmetrical if for all A, B ⊂ N, h(A) = h(B) whenever |A∩N_i| = |B∩N_i|, i = 1,..., t. The set of all the p-symmetrical entropy functions, denoted by ψ_p*, is called p-symmetrical entropy function region. We prove that ψ_p*, the closure of ψ_p*, is completely characterized by Shannon-type information inequalities if and only if p is the 1-partition or a 2-partition with one of its blocks being a singleton. The characterization of the partition-symmetrical entropy functions can be useful for solving some information theory and related problems where symmetry exists in the structure of the problems.