Moreover, the ordinary generating function of these partitions is given by
Proposition 4.5 The ordinary generating function for the reduced permutations of genus 1, counting the number of points and cycles, is given by:
Thus the radius of convergence of the ordinary generating function of K is [[rho].sub.U] = 3[square root of 2].
Let K be an adequate class with unlabelled count functions and ordinary generating functions as described above.
We show that the ordinary generating function for unlabelled (3 + 1)-free posets is
where [B.sub.unl](x) = 1 + 2x + 4[x.sup.2] + 8[x.sup.3] + 17[x.sup.4] + *** is the ordinary generating function for unlabelled bicoloured graphs.
The transfer matrix approach Marberg used to establish the rationality of the ordinary generating function, [[summation].sub.n [greater than or equal to] 0] NC[N.sub.j,k] (n + 1, r)[x.sup.n] for set partitions of size n +1 is through translating the original problem to counting all closed walks of n-steps with certain column and row length restrictions (according to j, k) for each component from [empty set] [member of] [Y.sup.r], that is, r copies of the Hasse diagram of the Young lattice.
Most recently, Marberg (10) extended the result to coloured set partitions with a novel way of proving that the ordinary generating functions of j- noncrossing, k-nonnesting, r-coloured partitions according to size n are rational functions.
It is enough to find the ordinary generating function. Specifically, we define
In Section 3, we describe the transformation T that turns certain ordinary generating functions into Laguerre series.
For unlabeled structures, the (ordinary) generating function of C is C(z) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Since each element of F is obtained by taking a multiset of elements of C, the ordinary generating function
of F is given by (see )
Let us remark that the reason for the use of exponential generating functions, rather than ordinary generating functions
(o.g.f.), is simplicity of transfer rules in the domain of labelled classes.