A weak composition of a nonnegative integer is an ordered tuple of nonnegative integers whose sum is . It differs from a composition, whose parts must be positive. For example, and are distinct weak compositions of 3 (Tesler 2019).

For a positive integer , the number of weak compositions of into parts is

To count them, arrange stars and separators in a row. The numbers of stars before, between, and after the separators are the ordered parts. Choosing the separator positions gives the binomial coefficient above. For , there is one weak composition of each positive length, consisting entirely of zeros. There is also one empty weak composition of 0 (Tesler 2019).