Editing Exponential formula (section)

==Examples==
* <math>b_3 = B_3(a_1,a_2,a_3) = a_3 + 3a_2 a_1 + a_1^3,</math> because there is one partition of the set <math>\{1,2,3\}</math> that has a single block of size <math>3</math>, there are three partitions of <math>\{1,2,3\}</math> that split it into a block of size <math>2</math> and a block of size <math>1</math>, and there is one partition of <math>\{1,2,3\}</math> that splits it into three blocks of size <math>1</math>. This also follows from <math>Z_3 (a_1,a_2,a_3) = {1 \over 6}(2 a_3 + 3 a_1 a_2 + a_1^3) = {1 \over 6} B_3 (a_1, a_2, 2 a_3) </math>, since one can write the [[group (mathematics)|group]] <math>S_3</math> as <math>S_3 = \{ (1)(2)(3), (1)(23), (2)(13), (3)(12), (123), (132) \}</math>, using cyclic notation for [[permutation]]s.
* If <math>b_n = 2^{n(n-1)/2}</math> is the number of [[graph (discrete mathematics)|graphs]] whose vertices are a given <math>n</math>-point set, then <math>a_n</math> is the number of [[connectivity (graph theory)|connected graphs]] whose vertices are a given <math>n</math>-point set.
* There are numerous variations of the previous example where the graph has certain properties: for example, if <math>b_n</math> counts graphs without cycles, then <math>a_n</math> counts [[tree (graph theory)|trees]] (connected graphs without cycles).
* If <math>b_n</math> counts [[directed graph]]s whose {{em|edges}} (rather than vertices) are a given <math>n</math> point set, then <math>a_n</math> counts connected directed graphs with this edge set.
*In [[quantum field theory]] and [[statistical mechanics]], the [[Partition function (mathematics)|partition functions]] <math>Z</math>, or more generally [[correlation function]]s, are given by a formal sum over [[Feynman diagram]]s. The exponential formula shows that <math>\ln(Z)</math> can be written as a sum over connected Feynman diagrams, in terms of [[connected correlation function]]s.