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

• Published in: Discrete Applied Mathematics Vol. 272; pp. 90 - 99
• Main Authors: Turcu, Flavius, Bonchiş, Cosmin, Najim, Mohamed
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.01.2020; Elsevier BV; Elsevier
• Subjects: Algorithms; Classical Analysis and ODEs; Combinatorial analysis; Computer Science; Discrete Mathematics; Mathematical analysis; Mathematics; Multi-dimensional Bell polynomials; Multi-dimensional Faà di Bruno formulae; Partitions (mathematics); Polynomials; Vector partitions; Multi-dimensional Bell polynomials; Vector partitions; Multi-dimensional Faà di Bruno formulae
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2018.09.012

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.