Weak composition of an integer ($x_1+\cdots +x_k=N,x_i\ge 0$)

A weak composition of an integer $N\ge 0$ into $k\ge 1$ parts is an ordered tuple of nonnegative integers with sum $N$. Zero parts retain their positions: $(0,2,3)$ and $(2,0,3)$ differ. Fixing the length $k$ is essential when applying stars and bars, since inserting extra zero parts creates further tuples. ^[ref]