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Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named in honor of Francesco Faà di Bruno (1825–1888), who was (in chronological order) a military officer, a mathematician, and a priest, and was (posthumously, of course), beatified by the Pope. It can be stated in a general and perhaps initially forbidding form thus:
where
This may initially seem forbidding, so let us examine a concrete case, and see what the pattern is:
What is the pattern?
The factor g ′′ (x) g′ (x)2 corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factor f ′′′(x) that goes with it corresponds to the fact that there are three summands in that partition. The coefficient 6 that goes with those factors corresponds to the fact that there are exactly six partitions of a set of four members that break it into one part of size 2 and two parts of size 1.
Similarly for the other terms. That is the pattern.
In the formal power series
regardless of the bounds of summation (i.e. regardless of whether k runs from 0 to ∞ or from 1 to ∞, etc.), we have the nth derivative at 0:
This should not be construed as the value of a function, since these series are purely formal; there is no such thing as convergence or divergence in this context.
If
and
and
then the coefficient cn (which would be the nth derivative of h evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by
where π runs through the set of all partitions of the set { 1, ..., n } and B1, ..., Bk are the blocks of the partition π, and | Bj | is the number of members of the jth block, for j = 1, ..., k.
This version of the formula is particularly well suited to the purposes of combinatorics. See the "compositional formula" in Chapter 5 of Enumerative Combinatorics, Volumes 1 and 2, Richard P. Stanley, Cambridge University Press, 1997 and 1999, BooksEnthsiast.com.
If f(x) = ex then all of the derivatives of f are the same, and are a factor common to every term. In case g(x) is a cumulant-generating function, then f(g(x))) is a moment-generating function, and the polynomial in various derivatives of g is the polynomial that expresses the moments as functions of the cumulants.
These partition-counting Faà di Bruno coefficients have a "closed-form" expression. The number of partitions of a set of size n corresponding to the integer partition
of the integer n is equal to
These coefficients also arise in the Bell polynomials, which are relevant to the study of cumulants.
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