
32 Polytope Projects
The relation between the pointing of species and the corresponding
series is summarized by the following equalities:
)x(F
dx
d
x)x(F =
∗
(2.29)
,...)x,x,x(Z
x
x)x(F
~
32
F
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=
∗
(2.30)
,...)x,x(Z
x
,...)x,x(Z
21F
1
21
F
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=
∗
(2.31)
Figure 2.13 shows possible representations of the integral ∫F (Leroux
and Viennot 1988a).
Fig. 2.13 Representations of the integral ∫F.
=
F
2
3
4
● 1
1
∫F
4
3
2
To satisfy combinatorial needs, variations of the notion of combinatorial
species have been introduced.
In enumerative combinatorics, it is often required to consider some
parameters related to the characteristics of the structures. For instance,
the ...