Vector partitions, multi-dimensional Faà di Bruno formulae and generating algorithms

Abstract

In the paper we introduce and study partitions of vectors in Np, as a natural extension of the classical partitions of integers. We show that these vector partitions are the suitable combinatorial notion in extending the Faà di Bruno formula to a general multi-variable setting, as well as in generalizing Bell polynomials. The Adomian polynomials can be obtained from this framework as a particular case. We also give a recursive algorithm which is proved to generate all the vector partitions without repetitions, and which can be used in numerical applications of extended Faà di Bruno formulae and generalized Bell polynomials.