On the number of multiplicative partitions of a multi-partite number (Q2744385)

Let \(f_t(n_1,n_2, \dots,n_t)\) denote the number of multiplicative partitions of \((n_1,n_2, \dots,n_t)\). Here \(n_1,n_2,\dots,n_t\) are positive integers and multiplication is performed coordinatewise. For example, \(f_3(4,3,2)=11\) because NEWLINE\[NEWLINE\begin{aligned} (4,3,2) & =(4,3,1)(1,1,2)=(4,1,2)(1,3,1)\\ & =(4,1,1)(1,3,2)= (4,1,1)(1,3,1)(1,1,2)\\ & =(2,3,2)(2,1,1)= (2,3,1)(2,1,2)= (2,3,1)(2,1,1) (1,1,2)\\& =(2,1,2)(2,1,1)(1,3,1)= (2,1,1)(2,1,1)(1,3,2)\\ & =(2,1,1)(2,1,1) (1,3,1)(1,1,2).\end{aligned}NEWLINE\]NEWLINE The authors prove that NEWLINE\[NEWLINEf_t(n_1,n_2, \dots,n_t) \leq M^{w(t)},\tag{1}NEWLINE\]NEWLINE where \(M=n_1n_2\dots n_t\) and \(w(t)= {\log((t+1)!)\over t\log 2}\). When \(t=1\), this result reduces to \(f_1(n)\leq n\). However, as the authors point out, it is known that \(f_1(n)\leq{n\over\log n}\) for all \(n>1\), \(n\neq 144\). Therefore possibly the formula (1) may yet be improved further.