Ordered Factorizations with $k$ Factors

Abstract

We give an overview of combinatoric properties of the number of ordered $k$-factorizations $f_k(n,l)$ of an integer, where every factor is greater or equal to $l$. We show that for a large number $k$ of factors, the value of the cumulative sum $F_k(x,l)={\sum }_{n\le x}{f}_k(n,l)$ is a polynomial in $\lfloor {\log }_{l}x\rfloor$ and give explicit expressions for the degree and the coefficients of this polynomial. An average order of the number of ordered factorizations for a fixed number $k$ of factors greater or equal to 2 is derived from known results of the divisor problem.