EXPONENTIAL FORMULA

In combinatorial mathematics, the 'exponential formula' states that for any formal power series of the form
:$f(x)=a\_1\; x+\{a\_2\; over\; 2\}x^2+\{a\_3\; over\; 6\}x^3+cdots+\{a\_n\; over\; n!\}x^n+cdots,$
we have
:$exp\; f(x)=e^\{f(x)\}=1+sum\_\{n=1\}^infty\; \{b\_n\; over\; n!\}x^n,,$
where
:$b\_n=sum\_\{pi=left\{,B\_1,,dots,,B\_k,$
ight}} a_{left|B_1
ight|}cdots a_{left|B_k
ight|},
and the index π runs through the list of all partitions { ''B''₁, ..., ''B''_''k'' } of the set { 1, ..., ''n'' }. For example,
:$b\_3=a\_3+3a\_2\; a\_1\; +\; a\_1^3,,$
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 ''a''₁, ''a''₂, ''a''₃ is a Bell polynomial.
Essentially, the exponential formula is a power-series version of a special case of Faà di Bruno's formula.

Contents

Bell polynomials

References

Bell polynomials

One can write the formula in the following form, where ''B''_''n''(''a''₁, ..., ''a''_''n'') is the ''n''th complete Bell polynomial:
:$expleft(sum\_\{n=1\}^infty\; \{a\_n\; over\; n!\}\; x^n$
ight)
=1 + sum_{n=0}^infty {B_n(a_1,dots,a_n) over n!} x^n.

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.