nLab Faa di Bruno formula

Faà di Bruno formula is a remarkable combinatorial formula for higher derivatives of a composition of functions. There are various modern approaches to the related mathematics, using Joyal’s theory of species, operads, graphs/trees, combinatorial Hopf algebras and so on.

Literature

• Imma Gálvez-Carrillo, Joachim Kock, Andrew Tonks, Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees arxiv/1207.6404 (cf. also Kock’s combinatorics page)

We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs.

• Doron Zeilberger, Toward a combinatorial proof of the Jacobian conjecture? in Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985), 370–380, Lecture Notes in Math. 1234, Springer 1986. MR89c:05009
• Eliahu Levy, Why do partitions occur in Faa di Bruno’s chain rule for higher derivatives?, math.GM/0602183.
• E. Di Nardo, G. Guarino, D. Senato, A new algorithm for computing the multivariate Faà di Bruno’s formula, arxiv/1012.6008
• Miguel A. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602

In works of T. J. Robinson the formula is treated in the context of vertex algebras, calculus with formal power series and in logarithmic calculus, as well as in a connection to the umbral calculus:

• Thomas J. Robinson, New perspectives on exponentiated derivations, the formal Taylor theorem, and Faà di Bruno’s formula, Proc.Conf.Vert.Op.Alg., Cont.Math. 497 (2009) 185-198 arxiv/0903.3391; Formal calculus and umbral calculus, Electronic Journal of Combinatorics, 17(1) (2010) R95 arxiv/0912.0961

Faà di Bruno Hopf algebra

• Christian Brouder, Alessandra Frabetti, Christian Krattenthaler, Non-commutative Hopf algebra of formal diffeomorphisms, Adv. Math. 200:2 (2006) 479-524 pdf

• Kurusch Ebrahimi-Fard, Frederic Patras, Exponential renormalization, Annales Henri Poincare 11:943-971,2010, arxiv/1003.1679 doi

Using Dyson’s identity for Green’s functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed.

• Hector Figueroa, Jose M. Gracia-Bondia, Joseph C. Varilly, Faà di Bruno Hopf algebras, article at Springer eom, math.CO/0508337

• Jean-Paul Bultel, Combinatorial properties of the noncommutative Faà di Bruno algebra, J. of Algebraic Combinatorics 38:243–273 (2013) MR3081645

We give a new combinatorial interpretation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder–Frabetti–Krattenthaler for the antipode of the noncommutative Faà di Bruno algebra.