Exponential formula
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In combinatorial mathematics, the exponential formula states that for any formal power series of the form
we have
where
and the index π runs through the list of all partitions { B1, ..., Bk } of the set { 1, ..., n }. For example,
because there is one partition of the set { 1, 2, 3 } that has a single block of size 3, there are three partitions of { 1, 2, 3 } that split it into a block of size 2 and a block of size 1, and there is one partition of { 1, 2, 3 } that splits it into three blocks of size 1. This polynomial in the three variables a1, a2, a3 is a Bell polynomial.
Essentially, the exponential formula is a power-series version of a special case of Faà di Bruno's formula.
[edit] Bell polynomials
One can write the formula in the following form, where Bn(a1, ..., an) is the nth complete Bell polynomial:
[edit] References
See Chapter 5 of Enumerative Combinatorics, Volumes 1 and 2, Richard P. Stanley, Cambridge University Press, 1997 and 1999, ISBN 0-521-55309-1N.
