Editing Combinatorial species (section)

===Differentiation===
'''[[derivative|Differentiation]]''' of species intuitively corresponds to building "structures with a hole", as shown in the illustration below.

[[File:Derivative of combinatorial species.svg|200px|center]]

Formally,<ref>{{harvnb|Joyal|1981|loc=§ 2.3. Definition 8}}</ref>

:<math>(F')[A] = F[A \uplus \{\star\}],</math>

where <math>\star</math> is some distinguished new element not present in <math>A</math>.

To differentiate the associated exponential series, the sequence of coefficients needs to be shifted one place to the "left" (losing the first term). This suggests a definition for species: ''F' ''[''A'']&nbsp;=&nbsp;''F''[''A''&nbsp;+&nbsp;{*}], where {*} is a singleton set and "+" is disjoint union. The more advanced parts of the theory of species use differentiation extensively, to construct and solve [[differential equation]]s on species and series. The idea of adding (or removing) a single part of a structure is a powerful one: it can be used to establish relationships between seemingly unconnected species.

For example, consider a structure of the species ''L'' of linear orders&mdash;lists of elements of the ground set. Removing an element of a list splits it into two parts (possibly empty); in symbols, this is ''L'''&nbsp;=&nbsp;''L''·''L''. The exponential generating function of ''L'' is ''L''(''x'')&nbsp;=&nbsp;1/(1&nbsp;−&nbsp;''x''), and indeed:

:<math>
  \frac d {dx} {(1-x)}^{-1} = {(1-x)}^{-2}.
</math>

The generalized differentiation formulas are to be found in a previous research by N. G. de Bruijn, published in 1964.

The species ''C'' of cyclic permutations takes a set ''A'' to the set of all cycles on ''A''. Removing a single element from a cycle reduces it to a list: ''C'''&nbsp;=&nbsp;''L''. We can [[Integral|integrate]] the generating function of ''L'' to produce that for&nbsp;''C''.

:<math>
  C(x) = 1 + \int_0^x \frac{dt}{1-t} = 1 + \log \frac{1}{1-x}.
</math>

A nice example of integration of a species is the completion of a line (coordinatizated by a field) with the infinite point and obtaining a projective line.