For an integer partition of \( n \) , the corresponding norm is defined as the product of its parts. For example, both partitions of \( 6 = 4 + 2 = 2 + 2 + 2 \) have the norm \( 8 \) . Although prior work has explored partition norms through infinite series, integral representations, asymptotic expressions, and recurrence relations, dedicated investigations into inequalities remain largely unexplored. In this study, we conduct a statistical analysis of partition norms by investigating their raw moments and entropy through indirect approaches that circumvent the issue of absence of a simple closed-form expression for the underlying distribution. A collection of inequalities is derived using tools such as the arithmetic–geometric mean inequality, the generalized abc conjecture, Jensen’s inequality, Levinson’s inequality, Bhatia–Davis inequality, Kantorovich’s inequality, and other information-theoretic inequalities. Formulating the problem as the study of raw moments also opens the door to more sophisticated statistical treatments.
