Counting rises, levels, and drops in compositions

الأصل · EN

A composition of n∈ is an ordered collection of one or more positive integers whose sum is n. The number of summands is called the number of parts of the composition. A palindromic composition of n is a composition of n in which the summands are the same in the given or in reverse order. In this paper we study the generating function for the number of compositions (respectively palindromic compositions) of n with m parts in a given set A with respect to the number of rises, levels, and drops. As a consequence, we derive all the previously known results for this kind of problem, as well as many new results.