  In 1979, building on S. Lie's theory of symmetries of (partial) differrential
equations, P.J. Olver formulated inductive formulas which are appropriate for
the computation of the prolongations of an infinitesimal Lie symmetry to jet
spaces, for an arbitrary number n\geq 1 of independent variables (x^1, ...,
x^n) and for an arbitrary number m \geq 1 of dependent variables (y^1, ...,
y^m). This paper is devoted to elaborate a formalism based on multiple
Kronecker symbols which enables one to handle these ``unmanageable''
prolongations and to discover the underlying complicated combinatorics.
Proceeding progressively, we write down closed explicit formulas in four cases:
n=m=1; n\geq 1, m=1; n=1, m\geq 1; general case n\geq 1, m\geq 1. As a subpart
of the obtained formulas, we recover four possible versions of the
(multivariate) Fa\`a di Bruno formula. We do not employ the classical formalism
based on the symmetric algebra ({\it cf.} e.g. H. Federer's book, p. 222),
because it hides several explicit sums in symbolic compactifications, and
because the presence of supplementary complexities ({\it e.g.} splitting of
indices, combinatorics of partial derivatives) impedes us to apply such
compactifications coherently. Our method of exposition is inductive: we conduct
our reasonings by analyzing several thoroughly organized formulas, by comparing
them together and by ``drifting'' towards generality, in homology with the
classical style of L. Euler.
