The Faa di bruno formula for one variable (Wikipedia) is
The combinatorial forms in terms of bell polynomials are also included
Similarly, the multivariate formula (Wikipedia) is expressed combinatorially.
A more generalized version of this multivariate formula is also found and proven in this article by L. Hernandez Encinas and J Munoz Masque
Few other citations have also proved that same formula, so I will refer to them as well.
https://www.ams.org/journals/tran/1996-348-02/S0002-9947-96-01501-2/S0002-9947-96-01501-2.pdf
https://arxiv.org/pdf/1012.6008.pdf
As the readers of these articles can see that in all these cases, Faa Di Bruno's formula was restricted to only two nested functions, there was no mentioning of extending the formula for several nested functions, which brings me back to my question that, What methodologies shall be opted in order to extend the multivariate Faa di Bruno's formula for several nested functions ?
Any advise or editing in the question format or any correction in the formatting would be acknowledged from my side as a part of learning.
Thanks for giving me your very precious time
Kabir Munjal
It is computationally difficult in order to combinatorially arrive at a compactified multivariate Faa di Bruno's formula for several nested function, however, attempts have been made by Samuel G.G Johnston of Kings College London along with Joscha Prochno in order to combinatorially arrive at a compactified form of the desired formula by introducing the notions of generations, and energy functions from Graph Theory which lead to important keys towards our answer, some applications have also been discussed such as a new involution formula for the inversion of multivariate power series, then this is used as a framework in order to understand the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobi conjecture, etc. The lth nested function is expressed as a map from R^n:----:R^n. I shall provide the link of this article mentioned above,
https://arxiv.org/pdf/1911.07458.pdf.
I shall suggest few more citations for the readers of this answer.
This article below by Aimin Xu and Chengjing Wang, also solves the same answer but in a more algebraic way as compared to the Johnston's article, discussing first about the roles of divided differences in Faa di Bruno formula, giving an explicit formula for the n-th divided difference of a multicomposite function, the relations between Bell Polynomials w.r.t to multicomposite functions is also established and also coming to the answer that I was looking for in the first place, that has also been combinatorially solved by the above article by Samuel GG Johnston. I shall provide the link for Xu's and Wang's
https://reader.elsevier.com/reader/sd/pii/S0898122109007561?token=40162D666448BF26A6CFBC08CAA45334584D07F2DCA5D538B72901613164B2D74F5DD13B167DC73B144CB5F223665A6C&originRegion=eu-west-1&originCreation=20230515123359
I shall also provide few more articles that give a higher horizon and build a strong insight towards Faa di Bruno's formula, it's versions and applications to fields like combinatorics, differential geometry, Advanced Calculus, Algebras and Cogebras,etc.,
https://downloads.hindawi.com/journals/ijmms/2000/498526.pdf.
https://hal.science/hal-00950525/document
https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=F0055262F51FD3569E0BD25B36259900?doi=10.1.1.214.1456&rep=rep1&type=pdf
https://arxiv.org/pdf/math/0601149.pdf