84
II. LABELLED STRUCTURES AND EGFS
P ROPOSITION II.4. Let
be the class of permutations with cycle lengths in
and with a number of cycles that belongs to
. The corresponding EGF
is
where
E XAMPLE 11.
I. 2. ADMISSIBLE CONSTRUCTIONS AND SPECIFICATIONS
9
with being a neutral structure (of size 0). (The neutral structure in this context plays a
role similar to that of the empty word in formal language
36
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
then yields
with
a Fibonacci number. In particular the number of words of length that do not
contain
is
, a quantity that grows at an expo
48
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
empty) subtrees. In other words, general trees with
binary trees with
nodes:
nodes are equinumerous with pruned
Graphically, this is illus
66
II. LABELLED STRUCTURES AND EGFS
where
has
are again assumed to be increasing with ranges Im
Im
. For instance, one
as seen by reduction of the left pair or, dually, by expansion of the right pair.
18
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
summary, general trees are enumerated by Catalan numbers:
where
is a Catalan number.
For this reason the term Catalan tree is often employ
60
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
One of the reasons for the revival of interest in combinatorial enumerations and properties of random structures is the analysis of algori
I. 3. INTEGER COMPOSITIONS AND PARTITIONS
I. 3.1. Compositions and partitions. First the definitions:
D EFINITION I.6. A composition of an integer is a sequence
integers (for some ) such that
A partit
6
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
We recognize here the formula for a product of two power series. Therefore, with
etc, one has
Thus in our terminology, the cartesian produc
42
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
The Lagrange Inversion Theorem precisely states that the expansion of an inverse function
(here ) are determined simply by coefficients of
30
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
A check is provided by computing the OGF corresponding to this new specification,
(28)
which reduces to
as it should.
The interest of the
I. 1. SYMBOLIC ENUMERATION METHODS
3
F IGURE 1.
The class
of all triangulations of regular polygons (with size defined as the number of triangles)
is a combinatorial class.
The counting sequence start
24
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
E XAMPLE 1. Compositions with restricted summands. In order to enumerate the class
of compositions of whose parts are only allowed to be t
Contents
Chapter I.
Combinatorial Structures and
Ordinary Generating Functions
I. 1. Symbolic enumeration methods
I. 2. Admissible constructions and specifications
I. 2.1. Basic constructions
I. 2.2.
II. 2. ADMISSIBLE LABELLED CONSTRUCTIONS
69
Constructible classes. Like in the previous chapter, we say that a class of labelled
objects is constructible if it admits a specification in terms of sums
78
II. LABELLED STRUCTURES AND EGFS
E XAMPLE 8. Random allocations (balls-in-bins model). Throw at random distinguishable balls into distinguishable bins. A particular realization is described by a wo
I. 4. WORDS AND REGULAR LANGUAGES
39
F IGURE 9. The 15 ways to partition a four-element domain into blocks
.
correspond to
Consider a set partition that is formed of blocks. Identify each block by its
54
I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS
An immediate consequence of admissibility theorems is the following proposition first
encountered by Chomsky and Schutzenberger [26] in th
ANALYTIC COMBINATORICS
SYMBOLIC COMBINATORICS
P HILIPPE F LAJOLET & ROBERT S EDGEWICK
Algorithms Project
INRIA Rocquencourt
78153 Le Chesnay
France
Department of Computer Science
Princeton University
90
II. LABELLED STRUCTURES AND EGFS
The enumerative result
is a famous one, attributed to the prolific British
mathematician Arthur Cayley (18211895) who had keen interest in combinatorial mathematics
I. 5. TREES AND TREE-LIKE STRUCTURES
51
These coincide with gambler ruin sequences, a familiar object from probability theory: a
player plays head and tails. He starts with no capital (
) at time 0; h
96
II. LABELLED STRUCTURES AND EGFS
In each case, refined estimates follow from a detailed analysis of corresponding generating functions, which is a main theme of [51] and especially [77]. Raw forms
II. 4. ALIGNMENTS, PERMUTATIONS, AND RELATED STRUCTURES
87
Like several other structures that we have been considering previously, permutation
allow for transparent connections between structural cons
I. 6. ADDITIONAL CONSTRUCTIONS
57
to which it suffices to apply Moebius inversion; see A PPENDIX : Arithmetical functions,
p. 165.
E XAMPLE 13. Indecomposable permutations. A permutation
(written here
I. 2. ADMISSIBLE CONSTRUCTIONS AND SPECIFICATIONS
15
I. 2.3. Constructibility and combinatorial specifications. In the framework just introduced, the class of all binary words is described by
where
th