Composition of an integer ($x_1+\cdots +x_k=N,x_i\ge 1$)

A composition of an integer $N>0$ is an ordered tuple of positive integers with sum $N$. Its length $k$ counts the parts. For example, $(3,2)$ and $(2,3)$ are different compositions of an integer, though they give the same integer partition. Stars and bars counts compositions of an integer with fixed length. ^[ref]