Log-Concavity and the Multiplicative Properties of Restricted Partition Functions

Source abstract

Abstract The partition function p ( n ) and many of its related restricted partition functions have recently shown independently to satisfy log-concavity: $$p(n)^2 \ge p(n-1)p(n+1)$$ p ( n ) 2 ≥ p ( n - 1 ) p ( n + 1 ) for $$n\ge 26$$ n ≥ 26 , and satisfy the inequality: $$p(n)p(m) \ge p(n+m)$$ p ( n ) p ( m ) ≥ p ( n + m ) for $$n\ge m\ge 2$$ n ≥ m ≥ 2 with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence $$\{x_n\}$$ { x n } satisfying a particular initial condition likewise satisfies the inequality $$x_nx_m \ge x_{n+m}$$ x n x m ≥ x n + m . This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation.