On the Bessenrodt-Ono type inequality for a wide class of A-partition functions

Abstract The A-partition function pA (n) enumerates those partitions of n whose parts belong to a fixed (finite or infinite) set A of positive integers. On the other hand, the extended A-partition function pA () is defined as an multiplicative extension of the A-partition function to a function on A-partitions. In this paper, we investigate the Bessenrodt-Ono type inequality for a wide class of A-partition functions. In particular, we examine the property for both the m-ary partition function bₘ (n) and the d-th power partition function pd (n). Moreover, we show that bₘ () (pd () ) takes its maximum value at an explicitly described set of m-ary partitions (power partitions), where is an m-ary partition (a power partition) of n. Additionally, we exhibit analogous results for the Fibonacci partition function and the 'factorial' partition function. It is worth pointing out that an elementary combinatorial reasoning plays a crucial role in our investigation.