| Partition function has always been an important research object in mathematics,which has important applications in subjects such as combinatorics,group theory,physics,representation theory and statistics.Generally speaking,partition function refers to additive partition function,which mainly studies its generating function,asymptotic distribution,arithmetic property and combinatorial meaning.Multiplicative partitions are also regarded as the content of enumerative combinatorics and conjoint analysis,and the study of its generating function,combinatorial property and maximum order also have their significance in combinatorics.In this dissertation,we mainly study the asymptotic mean and maximum order of two kinds of multiplicative partition functions.The main results achieved are as follows:First,we discuss the number of multiplicative partitions that decompose n into k-free factors,where factor order is not considered here.We use a series of algorithms such as the Dirichlet series to get the asymptotic mean of multiplicative partitions of k-free factors.According to the results obtained,we conjecture the asymptotic mean of the ordered multiplicative partitions of k-free factors.It is the number of multiplicative partitions that decompose n into k-free factors,where factor order is considered here.Second,we study the problem of maximum order of ordered multiplicative partitions of squarefree factors.We first prove the Dirichlet series and asymptotic mean of the ordered multiplicative partitions of squarefree factors.Then we use a series of algorithms such as Lagrange multiplier to calculate the maximum order of ordered multiplicative partitions of squarefree factors.Figure [0] Table [0] Reference [0]... |
