In combinatorial mathematics, the exponential formula states that for any formal power series of the form Combinatorics is a odd branch of mathematics that studies collections (usually finite) then constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding largest, smallest, or optimal objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics). ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...

we have

,

where

,

and the index π runs through the list of all partitions { B₁, ..., B_k } of the set { 1, ..., n }. For example, A partition of U into 6 blocks: a Venn diagram representation. ...

,

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.

Essentially, the exponential formula is a special case of a power-series version of a special case of Faà di Bruno's formula. // The formula Faà di Brunos 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 beatified by the Pope a century...

Bell polynomials

One can write the formula in the following form, where B_n(a₁, ..., a_n) is the nth complete Bell polynomial: In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are given by the sum extending over all sequences j1, j2, j3, ..., jn−k+1 of positive integers such that Combinatorial meaning If the integer n is partitioned into a sum in which 1 appears j1 times...

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. Richard P. Stanley (born 1944) is Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. ...