2024/01/29 by Gajdzica, Krystian
#11P82 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 05A17 #Secondary 05A20
paper · doi:10.48550/arxiv.2401.16267
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(\boldsymbolμ) 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(\boldsymbolμ) (pd(\boldsymbolμ)) takes its maximum value at an explicitly described set of m-ary partitions (power partitions), where \boldsymbolμ 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.