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{{Short description|Mathematical technique}}
{{About|the method in analytic combinatorics|the method in invariant theory|Symbolic method}}
In [[combinatorics]]
During two centuries, generating functions were popping up via the corresponding recurrences on their coefficients (as can be seen in the seminal works of [[Daniel Bernoulli|Bernoulli]], [[Leonhard Euler|Euler]], [[Arthur Cayley]], [[Ernst Schröder (mathematician)|Schröder]],
[[Srinivasa Ramanujan|Ramanujan]], [[John Riordan (mathematician)|Riordan]], [[Donald Knuth|Knuth]], {{ill|Louis Comtet|fr|lt=Comtet}}, etc.).
Bender and Goldman<ref>{{cite journal|last1=Bender|first1=E.A.|last2=Goldman|first2=J.R.|title=Enumerative uses of generating functions|journal=Indiana Univ. Math. J.|date=1971|volume=20|pages=753-764}}</ref>, Foata and Schützenberger<ref name="fs">{{cite journal|last1=Foata|first1=D.|last2=Schützenberger|first2=M.|title=Théorie géométrique des polynômes Eulériens|journal=Lectures Notes in Math.|date=1970|volume=138}}</ref>, and Joyal<ref>{{cite journal|last1=Joyal|first1=Andre|title=Une théorie combinatoire des séries formelles|journal=Adv. Math.|date=1981|volume=42|pages=1-82|ref=joy}}</ref>.▼
It was then slowly realized that the generating functions were capturing many other facets of the initial discrete combinatorial objects, and that this could be done in a more direct formal way: The recursive nature of some combinatorial structures
translates, via some isomorphisms, into noteworthy identities on the corresponding generating functions.
Following the works of [[George Pólya|Pólya]], further advances were thus done in this spirit in the 1970s with generic uses of languages for specifying combinatorial classes and their generating functions, as found in works by [[Dominique Foata|Foata]] and [[Marcel-Paul Schützenberger|Schützenberger]]<ref name="fs">{{cite book|last1=Foata|first1=Dominique|authorlink1=Dominique Foata|last2=Schützenberger|first2=Marcel-P.|authorlink2=Marcel-Paul Schützenberger|title=Théorie Géométrique des Polynômes Eulériens|series=Lecture Notes in Mathematics|date=1970|volume=138|doi=10.1007/BFb0060799|isbn=978-3-540-04927-2|doi-access=free|arxiv=math/0508232}}</ref> on permutations,
▲Bender and Goldman on prefabs,<ref>{{cite journal|last1=Bender|first1=
Note that this symbolic method in enumeration is unrelated to "Blissard's symbolic method", which is just another old name for [[umbral calculus]].
The symbolic method in combinatorics constitutes the first step of many analyses of combinatorial structures,
which can then lead to fast computation schemes, to asymptotic properties and [[Asymptotic distribution|limit laws]], to [[random generation]], all of them being suitable to automatization via [[computer algebra]].
== Classes of combinatorial structures ==
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:<math>\frac{g(z)^n}{|G|}.</math>
We are able to enumerate filled slot configurations using either
:<math> X^2/E_2 \; + \; X^3/E_3 </math>
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:<math> X/C_1 \; + \; X^2/C_2 \; + \; X^3/C_3 \; + \; X^4/C_4 \; + \cdots.</math>
Clearly we can assign meaning to any such power series of quotients (orbits) with respect to permutation groups, where we restrict the groups of degree ''n'' to the conjugacy classes <math>\operatorname{Cl}(S_n)</math> of the symmetric group <math>S_n</math>, which form a [[unique factorization ___domain]]. (The orbits with respect to two groups from the same conjugacy class are isomorphic.) This motivates the following definition.
A class <math>\mathcal{C}\in \mathbb{N}[\mathfrak{
:<math>\mathcal{C} = \sum_{n \ge 1} \sum_{G\in \operatorname{Cl}(S_n)} c_G (X^n/G)</math>
where <math>\mathfrak{
In the following we will simplify our notation a bit and write e.g.
:<math> E_2 + E_3 \text{
for the classes mentioned above.
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In the labelled case we have the additional requirement that ''X'' not contain elements of size zero. It will sometimes prove convenient to add one to <math>G(z)</math> to indicate the presence of one copy of the empty set. It is possible to assign meaning to both <math>\mathcal{C}\in\mathbb{Z}[\mathfrak{A}]</math> (the most common example is the case of unlabelled sets) and <math>\mathcal{C}\in\mathbb{Q}[\mathfrak{A}].</math> To prove the theorem simply apply PET (Pólya enumeration theorem) and the labelled enumeration theorem.
The power of this theorem lies in the fact that it makes it possible to construct operators on generating functions that represent combinatorial classes. A structural equation between combinatorial classes thus translates directly into an equation in the corresponding generating functions. Moreover, in the labelled case it is evident from the formula that we may replace <math>g(z)</math> by the atom ''z'' and compute the resulting operator, which may then be applied to EGFs. We now proceed to construct the most important operators. The reader may wish to compare with the data on the [[cycle index]] page.
=== The sequence operator
This operator corresponds to the class
:<math>L = \frac{1}{1 - X} = 1 +
and represents sequences, i.e. the slots are not being permuted and there is exactly one empty sequence. We have
:<math> F(z) = 1 + \sum_{n\ge 1} Z(
1 + \sum_{n\ge 1} f(z)^n = \frac{1}{1-f(z)}</math>
and
:<math> G(z) = 1 + \sum_{n\ge 1}
\frac{1}{1-g(z)}.</math>
=== The cycle operator
This operator corresponds to the class
:<math>C = C_1 + C_2 + C_3 + \cdots</math>
i.e., cycles containing at least one object. We have
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:<math>
F(z) = \sum_{n\ge 1} Z(C_n)(f(z), f(z^2), \ldots, f(z^n)) =
\sum_{n\ge 1} \frac{1}{n} \sum_{d
or
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\log \frac{1}{1-g(z)}.</math>
This operator, together with the set operator
The labelled even cycle operator
:<math>C_2 + C_4 + C_6 + \cdots</math>
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\frac{1}{2} \log \frac{1}{1-g(z)^2}.</math>
This implies that the labelled odd cycle operator
:<math>C_1 + C_3 + C_5 + \cdots</math>
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\frac{1}{2} \log \frac{1+g(z)}{1-g(z)}.</math>
=== The multiset/set operator
The series is
:<math>E = 1 +
i.e., the symmetric group <math>S_n = E_n</math> is applied to the
The unlabelled case is done using the function
:<math>M(f(z), y) = \sum_{n\ge 0} y^n Z(
so that
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Evaluating <math>M(f(z), 1)</math> we obtain
:<math> F(z) = \exp \left( \sum_{
For the labelled case we have
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\sum_{n\ge 0} \frac{g(z)^n}{n!} = \exp g(z).</math>
In the labelled case we denote the operator by
:<math> F(z) = \exp \left( \sum_{
==Procedure==
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:<math>\sum_{k=0}^n A_k B_{n-k}.</math>
Using the definition of the OGF and some [[elementary algebra]], we can show that
:<math>\mathcal{A} = \mathcal{B} \times \mathcal{C}</math> implies <math>A(z) = B(z) \cdot C(z).</math>
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Other important elementary constructions are:
*the ''cycle construction'' (<math>\mathfrak{C}\{\mathcal{B}\}</math>), like sequences except that cyclic rotations are not considered distinct
*''pointing'' (<math>\Theta\mathcal{B}</math>), in which each member of
*''substitution'' (<math>\mathcal{B} \circ \mathcal{C}</math>), in which each atom in a member of
The derivations for these constructions are too complicated to show here. Here are the results:
{| class="wikitable"
|-
! Construction
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:<math>P(z) = \exp \left ( I(z) + \frac{1}{2} I(z^{2}) + \frac{1}{3} I(z^{3}) + \cdots \right ). </math>
Unfortunately, there is no closed form for <math>P(z)</math>; however, the OGF can be used to derive a [[recurrence relation]], or
===Specification and specifiable classes===
The elementary constructions mentioned above allow us to define the notion of ''specification''. This specification allows us to use a set of recursive equations, with multiple combinatorial classes.
Formally, a specification for a set of combinatorial classes <math>(\mathcal A_1,\dots,\mathcal A_r)</math> is a set of <math>r</math> equations <math>\mathcal A_i=\Phi_i(\mathcal A_1,\dots,\mathcal A_r)</math>, where <math>\Phi_i</math> is an expression, whose atoms are <math>\mathcal E,\mathcal Z</math> and the <math>\mathcal A_i</math>'s, and whose operators are the elementary constructions listed above.
A class of combinatorial structures is said to be ''constructible'' or ''specifiable'' when it admits a specification.
For example, the set of trees whose leaves' depth is even (respectively, odd) can be defined using the specification with two classes <math>\mathcal A_\text{even}</math> and <math>\mathcal A_\text{odd}</math>. Those classes should satisfy the equation <math>\mathcal A_\text{odd}=\mathcal Z\times \operatorname{Seq}_{\ge1}\mathcal A_\text{even}</math> and <math>\mathcal A_\text{even} = \mathcal Z\times \operatorname{Seq}\mathcal A_\text{odd}</math>.
==Labelled structures==
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===Boxed product===
In labelled structures, the min-boxed product <math>\mathcal{A}_{\min} = \mathcal{B}^{\square}\star \mathcal{C}</math> is a variation of the original product which requires the element of <math>\mathcal{B}</math> in the product with the minimal label. Similarly, we can also define a max-boxed product, denoted by <math>\mathcal{A}_{\max} = \mathcal{B}^
:<math>A_{\min}(z)=A_{\max}(z)=\int^z_0 B'(t)C(t)\,dt.</math>
or equivalently,
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===Example===
An increasing Cayley tree is a labelled non-plane and rooted tree whose labels along any branch stemming from the root form an increasing sequence. Then, let <math>\mathcal{L}</math> be the class of such trees. The recursive specification is now <math>\mathcal{L}=\mathcal{Z}^\square\star \operatorname{SET}(\mathcal{L}).</math>
===Other elementary constructions===
The operators
CYC{{sub|even}},
▲:<math>
CYC{{sub|odd}},
SET{{sub|even}},}}
and
}}
▲</math>
represent cycles of even and odd length, and sets of even and odd cardinality.
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:<math> \operatorname{SET}(\operatorname{CYC}(\mathcal{Z}))</math>
is used to study unsigned [[Stirling numbers of the first kind]], and in the derivation of the [[Random permutation statistics|statistics of random permutations]]. A detailed examination of the [[exponential generating function]]s associated to Stirling numbers within symbolic combinatorics may be found on the page on [[Stirling numbers and exponential generating functions in symbolic combinatorics]].
==See also==
*[[Combinatorial species]]
* [[Analytic combinatorics]]
==References==
{{reflist}}
* François Bergeron, Gilbert Labelle, Pierre Leroux, ''Théorie des espèces et combinatoire des structures arborescentes'', LaCIM, Montréal (1994). English version: ''Combinatorial Species and Tree-like Structures'', Cambridge University Press (1998).
* Philippe Flajolet and Robert Sedgewick, ''[[Analytic Combinatorics]]'', Cambridge University Press (2009). (available online: http://algo.inria.fr/flajolet/Publications/book.pdf)
* Micha Hofri, ''Analysis of Algorithms: Computational Methods and Mathematical Tools'', Oxford University Press (1995).
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