# exponential generating function

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## exponential generating function

[‚eks·pə¦nen·chəl ¦jen·ə‚rād·iŋ ′fəŋk·shən]
(mathematics)
A function, G (x), corresponding to a sequence, a0, a1, …, where G (x) = a0+ (a1 x /1!) + (a2 x 2/2!) + ⋯.
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In order to use combinatorial description of (17) we interpret exp(xt) as the exponential generating function of the labelled class of sets Set(XT) (see [3, Sect.
Let [M.sub.i,k](x) be the exponential generating function of the numbers [m.sub.i,k](n).
Consider the bivariate exponential generating function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where the coefficient of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the number [b.sub.n,p] of basis permutations of length n in [B.sub.p].
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the exponential generating function for labelled bicoloured graphs.
k 0 1 2 3 4 5 [p.sub.k] = dim 1 2 6 20 76 312 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] k 6 7 8 9 10 (27) [p.sub.k] = dim 1384 6512 32400 168992 921184 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The sequence [rb.sub.k] is [?] Sequence A000898 and it is related to the number of symmetric diagrams [b.sub.k] in the Brauer algebra (21) by the binomial transform [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and thus has exponential generating function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(i) Define the exponential generating function E(z) by (4.1) (assuming it to exist at least for small [absolute value of z]).
which is the exponential generating function for the number of set partitions into blocks of size strictly bigger than 1.
The coefficients in this exponential generating function form the sequence A000262 in Sloane's Encyclopedia of Integer Sequences that counts sets of lists.
where [B.sub.2n] [member of] Q are the Bernoulli numbers generated by the exponential generating function
It remains only to obtain the exponential generating function of [[micro].sub.n](a, b,1) as a ratio of cosines.
Compton's method to show that a given adequate class of finite relational structures K has a labelled 0-1 law is to show that its exponential generating function [A.sub.L](x) = [SIGMA][a.sub.L](n)[x.sup.n]/n!
(ii) Let B(x, y) be the exponential generating function enumerating the graphs in C, where x marks the vertices, and y marks the edges.
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