Linköping Studies in Science and Technology.
Dissertations, No. 1544
The k-assignment Polytope,
Phylogenetic Trees,
and
Permutation Patterns
Jonna Gill
Division of Mathematics and Applied Mathematics
Department of Mathematics
Linköping University, SE–581 83 Linköping, Sweden
Linköping 2013
The k-assignement Polytope, Phylogenetic Trees, and Permutation Patterns
Jonna Gill
Matematiska institutionen
Linköpings universitet
SE-581 83 Linköping, Sweden
[email protected]
Linköping Studies in Science and Technology.
Dissertations, No 1544
ISBN 978-91-7519-510-0
ISSN 0345-7524
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-98263
Copyright © 2013 Jonna Gill, unless otherwise noted.
Printed by LiU-Tryck, Linköping, Sweden 2013
Abstract
In this thesis three combinatorial problems are studied in four papers.
In Paper 1 we study the structure of the k-assignment polytope, whose vertices are the m × n (0,1)-matrices with exactly k 1:s and at most one 1 in each row
and each column. This is a natural generalisation of the Birkhoff polytope and
many of the known properties of the Birkhoff polytope are generalised. A representation of the faces by certain bipartite graphs is given. This representation is
used to describe the properties of the polytope, such as a complete description
of the cover relation in the face poset of the polytope and an exact expression
for the diameter of its graph. An ear decomposition of these bipartite graphs is
constructed.
In Paper 2 we investigate the topology and combinatorics of a topological
space, called the edge-product space, that is generated by a set of edge-weighted
finite semi-labelled trees. This space arises by multiplying the weights of edges
on paths in trees, and is closely connected to tree-indexed Markov processes
in molecular evolutionary biology. In particular, by considering combinatorial
properties of the Tuffley poset of semi-labelled forests, we show that the edgeproduct space has a regular cell decomposition with face poset equal to the
Tuffley poset.
The elements of the Tuffley poset are called X-forests, where X is a finite
set of labels. A generating function of the X-forests with respect to natural
statistics is studied in Paper 3 and a closed formula is found. In addition, a
closed formula for the corresponding generating function of X-trees is found.
Singularity analysis is used on these formulas to find asymptotics for the number
of components, edges, and unlabelled nodes in X-forests and X-trees as | X | →
∞.
In Paper 4 permutation statistics counting occurrences of patterns are studied. Their expected values on a product of t permutations chosen randomly from
Γ ⊆ Sn , where Γ is a union of conjugacy classes, are considered. Hultman has
described a method for computing such an expected value, denoted EΓ (s, t), of
a statistic s, when Γ is a union of conjugacy classes of Sn . The only prerequisite
is that the mean of s over the conjugacy classes is written as a linear combination of irreducible characters of Sn . Therefore, the main focus of this paper is
to express the means of pattern-counting statistics as such linear combinations.
A procedure for calculating such expressions for statistics counting occurrences
of classical and vincular patterns of length 3 is developed, and is then used to
calculate all these expressions.
iii
Populärvetenskaplig sammanfattning
I denna avhandling, som består av fyra artiklar, studeras tre olika områden inom
diskret matematik.
I den första artikeln studeras en generalisering av en polytop som kallas Birkhoffpolytopen. En polytop är ett geometriskt objekt med hörn och platta sidor,
som t.ex. ett mjölkpaket, en pyramid eller ett A4-papper. Hur en polytop ser ut
bestäms av koordinaterna för dess hörn. En polytops diameter är längsta avståndet mellan två hörn, där avståndet mellan två hörn är det minsta antalet kanter
som förbinder hörnen. Vad en sida i en polytop är beskrivs bäst av ett exempel:
Ett mjölkpaket har åtta 0-dimensionella sidor (hörnen), tolv 1-dimensionella sidor (kanterna), sex 2-dimensionella sidor (det vi vanligtvis kallar sidor), och en
3-dimensionell sida (hela mjölkpaketet). Diametern av ett mjölkpaket är 3.
Birkhoffpolytopen är polytopen vars hörns koordinater i n2 dimensioner ges
av alla n × n-matriser med exakt en etta i varje rad och i varje kolonn, och nollor
i övrigt. Polytopen som studeras här betecknas M (m, n, k), och koordinaterna
i m × n dimensioner för dess hörn ges av alla m × n-matriser med k ettor och
resten nollor, varav högst en etta i varje rad och varje kolonn. Om k = m = n fås
Birkhoffpolytopen.
En graf består av ett antal noder (punkter) och bågar som förbinder dem.
En bipartit graf är en graf där man kan dela upp noderna i två mängder, så
att alla bågar går mellan den ena mängden och den andra. En beskrivning av
M(m, n, k):s sidor som en speciell sorts bipartita grafer ges. Denna beskrivning
används sedan för att härleda ett exakt uttryck för M (m, n, k):s diameter.
I den andra och tredje artikeln studeras en partiellt ordnad mängd (en mängd
där vissa, men inte alla, element går att jämföra). Den kallas Tuffleypomängden,
och den är relaterad till en matematisk modell för hur mutationer i DNA sker.
En graf kallas ett träd om den inte innehåller några cykler (man kan inte gå
runt i cirkel i den). En graf som består av flera träd kallas en skog. En X-skog,
där X är en mängd märken, är en skog där man placerat ut märkena från X på
noderna enligt vissa regler. Tuffleypomängden består av alla X-skogar, och en
X-skog sägs vara mindre än en annan X-skog om den förra kan fås från den
senare genom att ta bort eller dra ihop bågar på ett visst sätt.
Det visas att Tuffleypomängden har något som kallas en rekursiv koatomordning, och att detta medför mycket trevliga egenskaper hos den matematiska modellen för mutationer.
En genererande funktion för X-skogar (som räknar antalet X-skogar med avseende på antalet märken, träd, bågar och noder utan märken) undersöks och
beskrivs som en komplex funktion. Denna funktion analyseras sedan för att ge
t.ex. medelvärde och varians för antalet bågar, noder utan märken och träd för
X-skogar med n märken (då n är stort).
I den fjärde artikeln studeras funktioner som räknar förekomsten av vissa
permutationsmönster. En permutation av talen {1, 2, . . . , n} är en uppräkning av
v
vi
Populärvetenskaplig sammanfattning
talen i någon ordning. En permutation π av {1, 2, . . . , n} kan skrivas π1 π2 . . . πn .
Ett permutationsmönster är en sekvens p1 p2 . . . pk , där k är längden på mönstret,
och mönstret sägs förekomma i permutationen π om det finns en delsekvens
πi1 πi2 . . . πik i π med samma relativa ordning som p1 p2 . . . pk . T.ex. förekommer
mönstret 132 i permutationen 624531, eftersom delsekvensen 243 har samma
relativa ordning som 132 (minst, störst, mittemellan). Totalt sett förekommer
detta mönster två gånger i permutationen (som delsekvenserna 243 och 253).
Man kan dela upp alla permutationer av talen {1, 2, . . . , n} i vissa naturliga
grupper som kallas konjugatklasser. Det finns också en slags multiplikation av
permutationer, vars resultat är en ny permutation. En fråga man kan studera
är följande: Om man slumpmässigt väljer t stycken permutationer från samma
konjugatklass och multiplicerar dem, vad är det förväntade antalet förekomster av ett visst permutationsmönster i resultatet? Den frågan kan besvaras om
ett slags medelvärdesfunktion för antalet förekomster av mönstret skrivs i en
speciell bas.
En metod för att finna sådana uttryck för alla mönster av längd 3 samt för ett
slags generaliserade mönster av längd 3 (där man t.ex. kan kräva att delsekvensen ska börja med π1 ) beskrivs, och alla deras medelvärdesfunktioner skrivs i
den speciella basen.
Acknowledgements
First of all I would like to thank my original supervisor Svante Linusson for his
support and enthusiasm for the research problems. I would also like to thank
my second supervisor Olle Axling, who has taught me a lot about teaching.
Secondly, I would like to thank my present supervisor Jan Snellman for taking over the supervision when Svante Linusson left Linköping University for
KTH Royal Institute of Technology. I would also like to thank my present second supervisor Axel Hultman for all help and support, and especially for his
thorough reading of and many comments on this manuscript.
Thanks also to the Swedish Research Council which has been supporting a
part of my graduate studies.
I would also like to thank all wonderful colleagues at the Department of
Mathematics who have made these 12 years of Ph.D. studies so pleasant.
Finally I would like to thank Jesus Christ for giving me my talent and love
for mathematics, and my family for their massive support during my studies. I
thank my parents and parents-in-law for all their help with babysitting and other
things. I also thank my five children for their patience during my writing and
for making my life so marvellous. At last I thank my beloved husband Johan for
more than it is possible to mention.
vii
Contents
I Introduction
1
Introduction
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Basic theory of polytopes . . . . . . . . . . . . . . . . . . . .
4.2
Transportation polytopes and network flow polytopes . . .
4.3
The Birkhoff polytope . . . . . . . . . . . . . . . . . . . . . .
5
The edge-product space and its face poset . . . . . . . . . . . . . .
5.1
Finite CW complexes . . . . . . . . . . . . . . . . . . . . . . .
5.2
The edge-product space of phylogenetic trees . . . . . . . .
5.3
The Tuffley poset . . . . . . . . . . . . . . . . . . . . . . . . .
6
The Lambert W function . . . . . . . . . . . . . . . . . . . . . . . . .
7
Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
The group Sn of permutations . . . . . . . . . . . . . . . . .
7.2
Representations of Sn and their characters . . . . . . . . . .
7.3
Permutation statistics . . . . . . . . . . . . . . . . . . . . . .
8
Overview of the papers . . . . . . . . . . . . . . . . . . . . . . . . .
Paper 1: The k-assignment polytope . . . . . . . . . . . . . . . . . .
Paper 2: A regular decomposition of the edge-product space of
phylogenetic trees . . . . . . . . . . . . . . . . . . . . . . . .
Paper 3: A generating function for X-forests . . . . . . . . . . . . .
Paper 4: Pattern containment in random permutations . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
3
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x
Contents
II Papers
1 The k-assignment polytope
1
Introduction . . . . . . . . . . . . . . . . . . . . . . .
2
Some basic properties of the k-assignment polytope
3
Description of the face poset . . . . . . . . . . . . .
4
The diameter of M (m, n, k) . . . . . . . . . . . . . .
5
Ear decomposition . . . . . . . . . . . . . . . . . . .
Appendix: Proof of Lemma 5.8 . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
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2 A regular decomposition of the edge-product space of phylogenetic
trees
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Trees, forests and the Tuffley poset . . . . . . . . . . . . . . . . . .
3
Recursive coatom orderings . . . . . . . . . . . . . . . . . . . . . . .
3.1
Preliminaries and definitions . . . . . . . . . . . . . . . . . .
3.2
Outline of proof of Theorem 3.1 . . . . . . . . . . . . . . . .
3.3
There is a recursive coatom ordering for [0̂, Γ] . . . . . . . .
3.4
An example of a coatom ordering satisfying (V3) . . . . . .
4
The edge-product space is a regular cell complex . . . . . . . . . .
5
Proof of some combinatorial lemmas . . . . . . . . . . . . . . . . . .
5.1
Reformulation of (V1) with implications . . . . . . . . . . .
5.2
Common elements of [0̂, α1 ] and [0̂, α2 ] . . . . . . . . . . . .
5.3
A and B are compatible with (V3) . . . . . . . . . . . . . . .
5.4
The order of the coatoms near two vertices . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
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3 A generating function for X-forests
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2
A generating function for X-forests . . . . . . . . . . . .
3
Recurrence relations for the coefficients . . . . . . . . . .
4
The generating functions . . . . . . . . . . . . . . . . . . .
5
Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Statistics of generating functions . . . . . . . . . .
5.2
Preliminaries on singularity analysis . . . . . . .
5.3
Singular expansions of derivatives of F and T . .
5.4
Asymptotic statistics of the generating functions
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1
Computation of the constants p j in Lemma 5.11 .
A.2
The functions f used in Section 5.3 . . . . . . . .
79
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xi
Contents
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 Pattern containment in random permutations
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Mean statistics of 3-patterns . . . . . . . . . . . . . . . . . . . .
4
A procedure for calculating the means of vincular 3-patterns
5
Expected values . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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103
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126
Part I
Introduction
1
Introduction
1 Introduction
This thesis consists of four papers.
First some basic theory about the subjects in the papers will be introduced,
and then an overview of the papers follows.
2 Posets
In all the following it will be assumed that all posets are finite. The notations
and definitions in this section can be found in e.g. [26].
Definition 2.1. A partially ordered set (or poset) is a set S equipped with a relation
≤, satisfying the following:
(i) For all x ∈ S, x ≤ x. (reflexivity)
(ii) If x ≤ y and y ≤ z, then x ≤ z. (transitivity)
(iii) If x ≤ y and y ≤ x, then x = y. (antisymmetry)
That x ≤ y and x ̸= y is denoted x < y. The element y covers x if x < y and
there is no z ∈ S such that x < z < y. For elements x, y ∈ S with x ≤ y, the
interval between x and y is [ x, y] := {z ∈ S : x ≤ z ≤ y}.
The Hasse diagram of a finite poset S is the graph whose nodes are the elements of S, whose edges are the cover relations, and such that if x < y then the
node y is drawn above x. In Figures 1 and 2, the Hasse diagrams of posets with
different properties are drawn.
There is a special property of posets, sometimes called the diamond property,
sometimes called thinness. A poset has this property if the interval [ x, y] has
3
4
Introduction
exactly four elements for each pair x, y ∈ S such that there exists a z ∈ S that
covers x and is covered by y.
A poset S is bounded if there are a unique minimal element 0̂ ∈ S and a
unique maximal element 1̂ ∈ S such that 0̂ ≤ x ≤ 1̂ for all x ∈ S. It is possible to
add an artificial 0̂ and/or 1̂ if needed.
Figure 1: Pure poset with the diamond property, bounded poset, graded poset.
A chain C ⊆ S is a sequence x0 < x1 < · · · < xn . The length of the chain is
ℓ(C ) = n. The length of a poset S is ℓ(S) := max{ℓ(C ) : C is a chain of S}. A
chain in S is maximal if no element in S can be added to the chain. The poset S is
pure if all maximal chains in S have the same length, and graded if it in addition
has a 0̂ and a 1̂. In a pure poset S there is a unique integer-valued rank function
r on S such that r ( x ), the rank of x, is 0 if x is a minimal element of S, and
r (y) = r ( x ) + 1 if y covers x in S.
An upper bound of x and y in S is an element z ∈ S such that x, y ≤ z. A lower
bound of x and y is an element w ∈ S such that w ≤ x, y.
Definition 2.2. A poset is a lattice if every two elements x, y ∈ S have a unique
minimal upper bound, called the join x ∨ y, and a unique maximal lower bound,
called the meet x ∧ y.
It follows that all (finite) lattices must have a 0̂ and a 1̂. If S is a lattice, then
the minimal elements of S ∖ 0̂ are called atoms, and the maximal elements of
S ∖ 1̂ are called coatoms.
Figure 2: A graded lattice, a non-pure lattice, and a poset which is not a lattice.
3 Graphs
5
Definition 2.3. A graded poset S is said to admit a recursive coatom ordering if the
length of S is 1 or if the length of S is greater than 1 and there is an ordering
α1 , . . . , αt of its coatoms that satisfies the following two conditions:
(i) For all i < j and γ < αi , α j there is a k < j and an element β such that β is
covered by αk and α j and γ ≤ β.
(ii) For all j = 1, . . . , t, the interval [0̂, α j ] admits a recursive coatom ordering in
which the coatoms that come first in the ordering are those that are covered
by some αk where k < j.
Recursive coatom orderings and properties of posets having such orderings
are described in [6, Chapter 4.7].
3 Graphs
The nodes in a graph are often called vertices, but here they will be called nodes
to avoid confusion, since polytopes (described in next section) have another kind
of vertices. Most of the theory in this section can be found in [21].
The nodes in a graph G connected by an edge x are called the endpoints of
x, and they are said to be incident to x. Two edges sharing a node are called
adjacent. A path in G is a sequence of distinct adjacent edges which connect a
sequence of distinct nodes. If every two nodes of a graph G are joined by a path,
then G is connected. A cycle is a path with at least two edges, together with an
edge joining the first and the last node in the path.
Definition 3.1. The diameter of a graph G will be denoted δ( G ) and is defined as
the smallest number δ such that any two nodes in G can be connected by a path
with at most δ edges. If G is not connected, the diameter is defined to be ∞.
The sets of vertices and edges of a graph G are often denoted V ( G ) and E( G ),
respectively. The graph H is a subgraph of G if V ( H ) ⊆ V ( G ) and E( H ) ⊆ E( G ).
An edge-induced subgraph of a graph G consists of some of the edges of G and
their endpoints. The degree deg(v) of a vertex v ∈ V ( G ) is the number of edges
incident to v. A weighted graph associates a weight (usually a real number) with
every edge in the graph.
Definition 3.2. A graph without cycles is called a forest. If it is also connected,
it is called a tree. Vertices with degree 1 are called leaves. A binary tree is a tree
where all vertices except the leaves have degree 3.
Definition 3.3. A matching in a graph is a subset of its edges such that no two
edges share an endpoint. A perfect matching is a matching that matches all nodes
of the graph, i.e. all nodes are incident to exactly one edge.
Recall that a bipartite graph is a graph whose nodes can be divided into two
disjoint sets V1 and V2 , such that every edge connects a node in V1 to one in
6
Introduction
Figure 3: A forest, a graph not being a forest, a tree, and a binary tree.
V2 . The complete bipartite graph Km,n is the bipartite graph with |V1 | = m and
|V2 | = n, where there is an edge (v1 , v2 ) for each pair of nodes v1 ∈ V1 and
v2 ∈ V2 .
Definition 3.4. A bipartite graph G is said to be elementary if each edge of G lies
in some perfect matching of G.
Figure 4: Matching, perfect matching, elementary graph, not elementary graph (the
highlighted edge does not lie in any perfect matching).
In [21] an elementary graph is required to be connected, but that is not necessary here. Each component of an elementary graph here will be elementary
according to the original definition.
Now the concept of ear decompositions will be described. The following notation is used: If E1 and E2 are subsets of E( G ), then E1 + E2 denotes the subgraph
of G induced by E1 ∪ E2 .
Definition 3.5. Let x be an edge. Join its endpoints by a path E1 (not containing
x) of odd length. Then a sequence of bipartite graphs can be constructed as
follows: If Gs−1 = x + E1 + · · · + Es−1 has already been constructed, add a new
ear Es by picking any two nodes that are connected by an odd path in Gs−1
and joining them by an odd path Es having no other node (and no edge) in
common with Gs−1 . The decomposition Gs = x + E1 + · · · + Es will be called an
ear decomposition of Gs , and Ei will be called an ear (i = 1, . . . , s).
Theorem 3.6. A bipartite graph G is elementary if and only if each component of G has
an ear decomposition.
7
4 Polytopes
4 Polytopes
The following is mainly from [11] and [27]. Only convex polytopes will be
considered, so the word convex will often be omitted.
4.1 Basic theory of polytopes
Definition 4.1. A (convex) polytope is a subset P ⊆ Rd that is the convex hull of a
finite point set,
P = Conv(V1 , . . . , Vn )
for some V1 , . . . , Vn ∈ Rd
or, equivalently, a subset P ⊆ Rd that is a bounded intersection of half-spaces,
P = {x ∈ Rd : ci · x ≤ zi , i = 1, . . . , m}
for some ci ∈ Rd , zi ∈ R.
That P is bounded means that P does not contain a ray {x + ty : t ≥ 0} for
any y ̸= 0.
Points, lines, planes, and so forth, are affine subspaces of Rd (they are not
required to include the origin), also called flats. The affine hull of a finite point
set is the intersection of all affine flats that contain the set. The dimension of a
polytope is the dimension of its affine hull.
Definition 4.2. Let P ⊆ Rd be a convex polytope. A face of P is any set of the
form
F = P ∩ { x ∈ Rd : c · x = c0 }
where c · x ≤ c0 for all x ∈ P.
Since 0 · x ≤ 0 for all x ∈ P, P is a face of itself. The other faces, satisfying
F ⊂ P, are called proper faces. The empty set, ∅, is always a face of P since
0 · x ≤ 1 for all x ∈ P.
The faces of dimensions 0, 1, dim( P) − 2, and dim( P) − 1 are called vertices,
edges, ridges, and facets, respectively. The set of vertices is denoted vert( P).
Theorem 4.3. Every polytope is the convex hull of its vertices. If a polytope can be
written as the convex hull of a finite point set, then the set contains all the vertices of the
polytope.
Theorem 4.4. Let P ⊆ Rd be a polytope, and let V := vert( P). Suppose that F is a
face of P. Then the following statements are true.
(i) The face F is a polytope, with vert( F ) = F ∩ V.
(ii) Every intersection of faces of P is a face of P.
(iii) The faces of F are exactly the faces of P that are contained in F.
Definition 4.5. The face poset of a convex polytope P is the poset L( P) of all faces
of P, partially ordered by inclusion (i.e. the relation ’≤’ is ⊆).
8
Introduction
P
1245
3
123
135
234
345
P
5
12
1
13
4
5
2
1
15
24
23
2
3
4
34
35
4
45
5
3
1
2
Ø
Figure 5: A polytope P, its face lattice L( P), and its graph G ( P).
Theorem 4.6. Let P be a convex polytope. The face poset L( P) is a graded lattice of
length dim( P) + 1, with rank function r ( F ) = dim( F ) + 1. (Hence L( P) is often
called the face lattice of P.) The face lattice L( P) has the diamond property.
Definition 4.7. Let P be a convex polytope. The graph of P, denoted G ( P), is
the graph formed by the vertices and the edges of P. The diameter of P is the
diameter of its graph G ( P) and will be denoted δ( P).
4.2 Transportation polytopes and network flow polytopes
The transportation problem is a classic problem in optimisation. Suppose that a
product is to be transported from m warehouses to n customers. The i:th warehouse produces ri > 0 units of the product per time unit, and the j:th customer
requires a j > 0 units of the product per time unit. The cost for transporting one
unit of the product from the i:th warehouse to the j:th customer is cij , and the
number of units transported is xij . The goal is to minimise the total transportation cost. Hence the transportation problem of order m × n is to minimise the
linear function
m
n
∑ ∑ cij xij
i =1 j =1
subject to the conditions
n
∑ xij = ri ,
i = 1, . . . , m,
j =1
m
∑ xij = a j ,
j = 1, . . . , n,
i =1
xij ≥ 0,
i = 1, . . . , m, j = 1, . . . , n.
The set of matrices ( xij )m×n satisfying these conditions is called the transportation
polytope, usually denoted T (r, a).
9
4 Polytopes
Network flows are described in [1]. Let G = ( N, A) be a directed network
(directed graph) defined by a set N of nodes and a set A of directed edges. Each
edge (i, j) ∈ A has a capacity uij that denotes the maximum flow on the edge,
and a lower bound ℓij that denotes the minimum flow on the edge. Each node
has a number bi representing its supply/demand. An example of a graph of a
network flow is shown in Figure 6. The variable xij denotes the flow on the edge
(i, j) ∈ A. The set of possible solutions to
∑
{ j:(i,j)∈ A}
xij −
∑
{ j:( j,i )∈ A}
for all i ∈ N,
x ji = bi
ℓij ≤ xij ≤ uij
for all (i, j) ∈ A,
is a network flow polytope.
When minimising or maximising a linear function of the flows on the edges,
an optimal solution can always be found in a vertex of the network flow polytope.
b3
3
4
(l45 ,u45)
b4
2 −a2
r3
3
(0, )
8
(l24 ,u24)
1 −a1
8
b2
5 b5
(0, )
8
2
r2
(0, )
(0, )
2
(0, )
(0, )
(l35 ,u35)
(l43 ,u43)
(l34 ,u34)
1
8
(l12 ,u12)
r1
8
1
(l13 ,u13)
8
b1
Figure 6: A graph of a network flow, and the transportation problem of order 3 × 2
represented as a network flow.
It is rather obvious that all transportation polytopes are network flow polytopes. The m warehouses are represented by m supply nodes with bi = ri and the
n customers are represented by n demand nodes with b j = − a j . Each warehouse
has distribution channels to each customer, represented by directed edges from
the supply nodes to the demand nodes. All lower bounds are 0 and all capacities
are infinite. An example is given in Figure 6.
4.3 The Birkhoff polytope
The Birkhoff polytope has many names, such as the permutation polytope, the
assignment polytope, the polytope of doubly stochastic matrices, the perfect matching
polytope, and so forth. It can be defined using permutations. (Permutations and
the symmetric group Sd are described in Section 7.1.)
10
Introduction
Definition 4.8. For every permutation σ ∈ Sd , construct a d × d matrix X σ by
{
1 if σ (i ) = j
σ
Xij =
0 otherwise.
The matrices X σ are the 0/1-matrices with exactly one 1 in each row and exactly
one 1 in each column. They can be seen as 0/1-vectors in Rd×d , and their convex
hull forms a 0/1-polytope (a polytope where all vertex coordinates are 0 or 1)
called the Birkhoff polytope:
Bd := Conv{ X σ : σ ∈ Sd } ⊆ Rd×d .
The Birkhoff polytope Bd has d! vertices, d2 facets, and dimension (d − 1)2 .
Two vertices X σ and X π are the vertices of an edge if and only if the permutation
σ−1 π has exactly one cycle of length greater than 1. The diameter δ( Bd ) is 1 if
d ≤ 3, and 2 if d ≥ 4. The points in Bd are precisely
{
X∈R
d×d
: xij ≥ 0 for all i, j,
d
d
i =1
j =1
∑ xij = 1 for all j, ∑ xij = 1 for all i.
}
Hence the Birkhoff polytope Bd is the transportation polytope T (r, a) where
m = n = d, all ri = 1, and all a j = 1.
In [4], the following bijection between the faces of Bd and the elementary
graphs with 2d nodes is given. Every vertex V of Bd corresponds to a perfect
matching where the edge (i, j) is in the matching if and only if vij = 1. A face
of Bd corresponds to the elementary graph G that is the union of the perfect
matchings corresponding to the vertices of the face. If the face corresponding to
an elementary graph G is denoted F B ( G ), then the vertices of F B ( G ) are exactly
the vertices that correspond to all perfect matchings P such that P ⊆ G. In that
way the face poset of Bd is isomorphic to the lattice of all elementary subgraphs
of Kd,d ordered by inclusion.
Remember that all components of elementary graphs have ear decompositions. The following relationship between the dimension of a face in Bd and ear
decompositions of the corresponding graph is proved in [4].
Theorem 4.9. If G is an elementary bipartite graph, then the total number of ears in
ear decompositions of all the components of G is equal to the dimension of F B ( G ).
For more information about the Birkhoff polytope, see e.g. [4], [7], [8], and
[11].
5 The edge-product space and its face poset
5.1 Finite CW complexes
The following definitions can be found in any book on point set topology and
[5].
11
5 The edge-product space and its face poset
The closed d-ball Bd is defined to be the set { x ∈ Rd : | x | ≤ 1} (where | · | is
the standard norm in Rd ).
A topological space Y is a Hausdorff space if, for each x, y ∈ Y such that x ̸= y
there are disjoint open sets U, V with x ∈ U and y ∈ V.
If f : X → Y is bijective and f and f −1 are both continuous, f is called a
homeomorphism, and X and Y are said to be homeomorphic.
Let Y be a Hausdorff space. A subset σ is called an open d-cell if there exists
a continuous mapping ψ : Bd → Y whose restriction to the interior of the d-ball
is a homeomorphism ψ : Int Bd → σ. This defines the dimension dim σ = d
uniquely. The closure σ is the corresponding closed cell. In fact, σ = ψ( Bd ). Let
δ(σ ) denote the boundary σ ∖ σ.
Definition 5.1. Suppose that there is a finite collection C = {σα : α ∈ A}
of disjoint open cells whose union is Y, with corresponding maps ψα . The
space Y is a finite CW complex and the collection C is a cell decomposition of Y if
δ(σα ) ⊆ C <dim σα (the union of all cells in C of dimension less than dim σα ) for all
α ∈ A.
The face poset of a finite CW complex Y is the collection of closed cells σα
partially ordered by inclusion.
If each mapping ψα : Bdim σα → Y can be chosen to be a homeomorphism on
all of Bdim σα , then C is a regular cell decomposition of Y. An important property
of a regular CW complex Y is that it is homeomorphic to the so-called geometric
realisation of its face poset. This means that the topological properties of Y are
given by the combinatorial properties of its face poset.
5.2 The edge-product space of phylogenetic trees
The edge-product space of phylogenetic trees is defined in [22], as follows:
Take a finite set X. The binary trees T with the elements of X as their leaves
are all the possible “binary phylogenetic trees” for X. See [24]. If these trees are
given edge weights in the interval
( )[0, 1], they give rise to a space E ( X ) which
will now be described. (The set X2 denotes the set of all 2-element subsets of
X.)
Definition 5.2. Let λ be a map from E( T ) to [0, 1]. Define a new map p(T,λ) from
(X)
to [0, 1] by
2
p(T,λ) ( x, y) =
∏ λ ( e ),
e∈ P( T;x,y)
where P( T; x, y) is the set of edges in the path in T from x to y.
X
Let E ( X, T ) ⊂ [0, 1]( 2 ) denote the image of the map
X
Λ T : [0, 1] E(T ) → [0, 1]( 2 ) , λ 7→ p(T,λ)
X
and let E ( X ) denote the union of the subspaces E ( X, T ) of [0, 1]( 2 ) over all binary
trees T with X as their set of leaves. Then E ( X ) is called the edge-product space
for trees on X.
12
Introduction
In [22] it was shown that E ( X ) is a finite CW complex, and that the Tuffley
poset S( X ) is isomorphic to the face poset of E ( X ).
5.3 The Tuffley poset
The elements in the Tuffley poset are called X-forests. They and their partial
order relation will be defined here. More details about X-trees, X-forests and
the Tuffley poset can be found in [22] and [24].
Definition 5.3. An X-tree T is a pair ( T; ϕ), where T is a tree and ϕ : X → V ( T )
is a map with the property that all vertices of T of degree at most two belong to
ϕ( X ). The vertices in V ( T ) ∖ ϕ( X ) are called unlabelled.
Definition 5.4. An X-forest is a collection F = {( A, T A ) : A ∈ µ} where µ is a
set partition of X (a collection of disjoint subsets of X whose union is X) and T A
is an A-tree for each A ∈ µ.
4
2
1,3,8,11
5
1
6
4,7,10
10
5
7
9
11
6
2, 9
8
3
Figure 7: An X-tree and an X-forest, where X = {1, 2, . . . , 11}.
To remove an edge e = (u, v) from an X-forest T = ( T; ϕ) and identify u and
v, labelling the new vertex ϕ−1 (u) ∪ ϕ−1 (v), is called a contraction of the edge e.
Let S( X ) denote the set of X-forests. A partial order relation on S( X ) is
defined by F2 ≤ F1 if F2 can be obtained from F1 by contracting certain edges,
and deleting certain other edges, with any resulting unlabelled vertices of degree
2 being suppressed. The poset S( X ) is called the Tuffley poset on X.
Now the cell decomposition of E ( X ) given in [22] will be described. To
an X-tree T , associate the closed ‘cube’ B(T ) = [0, 1] E(T ) and the open ‘cube’
(
)
Int B(T ) = (0, 1) E(T ) . Then for an X-forest F = {( A, T A ) : A ∈ µ}, define
(
)
(
)
B(F ) = ∏ A∈µ B(T A ), so that Int B(F ) = ∏ A∈µ Int B(T A ) . The sets B(F )
(
)
and Int B(F ) are homeomorphic to a closed ball and an open ball, respectively,
of dimension ∑ A∈µ | E(T A )| (this quantity will be called the dimension of F ).
Given
( )an X-tree T = ( T; ϕ) and a map λ : E( T ) → [0, 1], define the map
p(T ,λ) : X2 → [0, 1] by
p(T ,λ) ( x, y) =
∏
e∈ P( T;ϕ( x ),ϕ(y))
λ ( e ),
(
)
where P T; ϕ( x ), ϕ(y) is the set of edges in the path in T from the node labelled
x to the node labelled y. If ϕ( x ) = ϕ(y), then p(T ,λ) ( x, y) := 1.
13
6 The Lambert W function
For an X-forest F = {( A, T A ) : A ∈ µ} and a map λ = (λ A , A ∈ µ), let
X
ψF : B(F ) → [0, 1]( 2 ) be defined by
{
p(T A ,λ A ) ( x, y)
ψF (λ)( x, y) =
0
if ∃ A ∈ µ such that x, y ∈ A,
otherwise.
The edge-product space E ( X ) is a CW complex with cell decomposition
{ (
)
}
ψF Int(B(F )) } : F ∈ S( X ) .
{ (
)
}
The face poset ψF B(F ) } : F ∈ S( X ) ordered by inclusion is isomorphic
to the Tuffley poset.
6 The Lambert W function
Definition 6.1. The Lambert W function (W (ζ )) is defined by W (ζ )eW (ζ ) = ζ.
There are several solutions to this equation, and they are the different branches
of the Lambert W function.
The derivative is given by
∂
∂ζ W ( ζ )
= (
W (ζ )
ζ 1 +W ( ζ )
) if ζ ̸= 0. The principal branch
W0 (ζ ) of the Lambert W function is the only branch defined at zero. It is analytic
in the whole complex plane except in the real interval ] − ∞, − 1e ], it is defined at
− 1e and real on the interval [− 1e , ∞[, and has derivative 1 at zero.
The Lambert W function is related to a generating function of trees. Let tn
be the number of rooted trees on n labelled nodes. The exponential generating
zn
function is T (z) = ∑∞
n=1 tn n! . It is known that the function T ( z ) = −W (− z ) and
that tn = nn−1 .
More about the Lambert W function can be found in [9] and [10].
7 Permutations
Most of the theory in this section can be found in [23] and [26].
7.1 The group Sn of permutations
Definition 7.1. A permutation of the set [n] := {1, 2, . . . , n} is a linear ordering
π1 π2 . . . πn of the elements of [n]. The expression π1 π2 . . . πn is called a word,
and the elements πi are consequently called letters. The permutation π can also
be seen as a bijective function π : [n] → [n] given by π (i ) = πi .
The inverse π −1 of a permutation π = π1 π2 . . . πn is given by π −1 (πi ) = i,
and the product of two permutations π and τ is defined to be the composition of
them as functions — that is (πτ )(i ) = π (τ (i )).
14
Introduction
The permutations of [n] with the operation multiplication form a group. This
group is called the symmetric group and will be denoted Sn .
Permutations can also be written in cycle notation. If (i1 i2 i3 . . . ik ) is a cycle
of π, then π (i1 ) = i2 , π (i2 ) = i3 , . . . , π (ik−1 ) = ik , and π (ik ) = i1 . Obviously,
this cycle can be regarded as identical to the cycle (i2 i3 . . . ik i1 ). It is easy to
see that every element of [n] appears in a unique cycle of π.
As an example, if π ∈ S7 is written as the word 3 2 1 7 5 4 6, then it can be
written as (2)(6 4 7)(5)(3 1) in cycle notation.
Definition 7.2. A partition of a positive integer n is a sequence of positive integers
λ = (λ1 , λ2 , . . . , λk ) such that ∑kj=1 λ j = n and λ1 ≥ λ2 ≥ · · · ≥ λk . That λ is a
partition of n is denoted λ ⊢ n.
The sequence of the lengths of all cycles in π in weakly decreasing order is
called the cycle type of π. Hence the cycle types of permutations in Sn are the
partitions of n. The permutation in the example above has cycle type (3, 2, 1, 1).
Two permutations are conjugate if and only if they have the same cycle
type.
Suppose that π,
(
) σ ∈ Sn and that (i1 i2 . . . ik ) is( a cycle
) of (π. Then
)
σ (i1 ) σ (i2 ) . . . σ (ik ) is a cycle of σπσ−1 , since (σπσ−1 ) σ(i1 ) = σ π (i1 ) =
σ (i2 ) and so on. Thus there is a bijection between partitions of n and conjugacy
classes of Sn .
Definition 7.3. A class function on Sn is a function from Sn to C that is constant
on conjugacy classes. A class function can also be seen as a function from the
partitions of n to C.
7.2 Representations of Sn and their characters
Now the representations of Sn and their characters will be briefly described. For
the full details, see [23]. Let GLd denote all invertible d × d matrices of complex
numbers, and let id denote the identity permutation, i.e. id = 1 2 . . . n.
Definition 7.4. A (matrix) representation of Sn is a map ρ : Sn → GLd such that
ρ(id) = Id (the identity matrix), and ρ(πτ ) = ρ(π )ρ(τ ) for all π, τ ∈ Sn . The
character of the representation ρ is the map χ : Sn → C which is defined by
χ(π ) = tr ρ(π ), where tr denotes the trace of a matrix.
Two matrix representations ρ1 and ρ2 of Sn are equivalent if and only if there
exists a fixed matrix T such that ρ2 (π ) = Tρ1 (π ) T −1 for all π ∈ Sn . In that case
their characters are identical.
It is easy to see that all characters of representations of Sn are class functions,
since χ(σπσ−1 ) = tr ρ(σπσ−1 ) = tr ρ(σ )ρ(π )ρ(σ)−1 = tr ρ(π ) = χ(π ).
Some of the representations of Sn are called irreducible representations (see
for example [23] for the exact definition). All the other representations, which
are said to be reducible, can be constructed using the irreducible representations
as building blocks. The characters of the irreducible representations are called
irreducible characters.
15
7 Permutations
There are as many irreducible representations of Sn as there are conjugacy
classes, and there is a standard bijection between them. The irreducible representation corresponding to the conjugacy class with cycle type λ will be denoted
ρλ , and the character of that representation will be denoted χλ .
Definition 7.5. Let f and g be any two functions from Sn to C. The inner product
of f and g is
1
f (π ) g(π )∗ ,
⟨ f , g⟩ =
n! π∑
∈S
n
where g(π )∗ is the complex conjugate of g(π ).
Theorem 7.6. The irreducible characters of Sn form an orthonormal basis for the space
of all class functions on Sn , with respect to the inner product defined above.
7.3 Permutation statistics
There are many interesting statistics on permutations, and some of them will be
described here. See for example [3].
Definition 7.7. A function s : Sn → N is called a permutation statistic.
The following definition from [18] constructs a class function s from any
permutation statistic s. If s is a class function, then s = s.
Definition 7.8. The mean statistic s is the class function which computes the mean
of s over conjugacy classes. If Cλ is the conjugacy class with cycle type λ, then
s(λ) =
1
s ( π ).
|Cλ | π∑
∈C
λ
By Theorem 7.6, every mean statistic s can be written as a linear combination
of irreducible characters.
Here follow some examples of common permutation statistics, and their
means will be expressed in the basis of irreducible characters. Suppose that
π = π1 π2 . . . πn is a permutation in Sn :
• The index i is a descent of π if πi > πi+1 . The permutation statistic counting
the number of descents of π is often denoted des(π ). The mean can be
1 (n)
written as des = n−
− n1 χ(n−1,1) − n1 χ(n−2,1,1) (see [16]).
2 χ
• The index i is an ascent of π if πi < πi+1 . The number of ascents of
π, asc(π ), is given by n − 1 − des(π ) = (n − 1)χ(n) (π ) − des(π ). Hence
1 (n)
asc = n−
+ n1 χ(n−1,1) + n1 χ(n−2,1,1) .
2 χ
• The pair (πi , π j ) is an inversion of π if i < j and πi > π j . The mean of the
number of inversions is inv =
[18]).
n ( n −1) ( n )
χ
4
−
n+1 (n−1,1)
6 χ
− 61 χ(n−2,1,1) (see
16
Introduction
• A fixed point of π is an index i such that πi = i. The number of fixed points
of π is the number of 1-cycles, so this is a class function and can be written
as χ(n) + χ(n−1,1) (see [19]).
Definition 7.9. A permutation π that swaps two elements is called a transposition. This means that π (i ) = i for all i ∈ [n] except two, call them i1 and i2 , for
which π (i1 ) = i2 and π (i2 ) = i1 . If i2 and i1 are adjacent, then π is called an
adjacent transposition.
The set of all transpositions is the conjugacy class with cycle type (2, 1, . . . , 1).
Definition 7.10. An occurrence of a (classical) pattern ϕ = ϕ1 ϕ2 . . . ϕk ∈ Sk in a
permutation π = π1 π2 . . . πn ∈ Sn is a subsequence in π of length k whose
letters are in the same relative order as those in ϕ.
For example, an occurrence of the classical pattern 123 in π ∈ Sn is a subsequence πi1 πi2 πi3 , where i1 < i2 < i3 , such that πi1 < πi2 < πi3 .
A generalisation of permutation patterns was described in [2]. Such patterns
are now called vincular patterns and are defined as follows:
Definition 7.11. A vincular pattern ϕ is written as a permutation in Sk enclosed by
brackets which may have dashes between adjacent letters. If two adjacent letters
are not separated by a dash, then the corresponding letters in an occurrence
of ϕ in π ∈ Sn must be adjacent. If ϕ begins with a square bracket then any
occurrence of ϕ in π must begin with π1 , and if ϕ ends with a square bracket
then any occurrence of ϕ in π must end with πn .
As an example, an occurrence of the vincular pattern [1-23) in π ∈ Sn is a
subsequence πi1 πi2 πi3 , where i1 < i2 < i3 , i1 = 1, and i3 = i2 + 1, such that
π i1 < π i2 < π i3 .
Let the number of occurrences of a pattern ϕ in π be denoted patϕ (π ). Many
permutation statistics can be written as sums of statistics counting occurrences
of vincular patterns. Consider for example the following statistic: A letter πi in
a permutation π is a peak if πi−1 < πi > πi+1 . If peak(π ) is the number of peaks
in π, it is easy to see that peak(π ) = pat(132) (π ) + pat(231) (π ).
Definition 7.12. If a permutation statistic s has the same distribution on Sn as
inv, then s is called Mahonian. That is, s is Mahonian if for all integers n ≥ 1
and k ≥ 0 the number of permutations π ∈ Sn with inv(π ) = k is equal to the
number of permutations τ ∈ Sn with s(τ ) = k.
A pattern function is a linear combination of statistics that count occurrences
of patterns. If each of these patterns have length at most d, then the pattern
function is called a d-function.
In [2] it is shown that many of the known Mahonian permutation statistics
are pattern functions, and all Mahonian 3-functions (up to some simple equivalences) are listed as such linear combinations.
8 Overview of the papers
17
8 Overview of the papers
Paper 1: The k-assignment polytope
The first paper is a study of a polytope called the k-assignment polytope. This
polytope is a generalisation of the well-known Birkhoff polytope Bn which has
the n × n permutation matrices as its vertices.
A natural generalisation of permutation matrices occurring both in optimisation and in theoretical combinatorics comes from k-assignments. A k-assignment
is k entries in a matrix that are required to be in different rows and columns. This
can also be described as placing k non-attacking rooks on a chess-board. The
k-assignment polytope M(m, n, k) is defined to be the polytope whose vertices are
the m × n (0, 1)-matrices with exactly k 1:s, and at most one 1 in each row and
each column.
In this paper, a description of the points in M(m, n, k) is given, and M(m, n, k)
is also described as a facet of a transportation polytope, and as a projection of
a network flow polytope. It is indicated how the description as a network flow
polytope can be used for linear optimisation over M(m, n, k).
The face poset of M(m, n, k) is investigated. A representation of the faces as
certain bipartite graphs, here called doped elementary graphs, is given. (There is an
equivalent representation of the faces as certain (0, 1)-matrices.) The representation as doped elementary graphs is used to describe the cover relation in the face
lattice of the polytope, and to give an exact expression for the diameter, which
turns out to be 1 when m, n ≤ (k + 1) if (m + n −(k ) ≤ 3 and 2 if (m) + n − k) ≥ 4.
If max(m, n) ≥ (k + 2), then the diameter is min max(m, n) − k, k .
Finally the concept of ear decompositions of bipartite graphs is generalised
to fit this problem, and an ear decomposition of the doped elementary graphs
is constructed. It is shown how this decomposition can be used to compute the
dimensions of the faces of M(m, n, k ).
This paper is a joint work with Svante Linusson. It is published in Discrete
Optimization, volume 6 (2009), pages 148–161.
Paper 2: A regular decomposition of the edge-product space of
phylogenetic trees
The second paper studies the edge-product space E ( X ) for trees on X, where X
is a fixed finite set.
One reason for investigating these spaces is that they are closely connected
to tree-indexed Markov processes in molecular evolutionary biology, see [22]. In
[22] it was shown that E ( X ) has a natural CW complex structure for any finite
set X, and a combinatorial description of the associated face poset was given.
This combinatorial description is the Tuffley poset S( X ) of X-forests.
In this paper it is shown that the edge-product space is a regular cell complex.
Here is an outline of the proof:
(
)
Using the notation in Section 5, it remains to show that the set ψF B(F )
is homeomorphic to [0, 1]dim F for all F ∈ S( X ). First it is concluded that it
18
Introduction
is enough to show the above for all X-trees T . Then
induction
on dim T is
(
)
used, with the induction hypothesis that the set ψF B(F ) is homeomorphic to
[0, 1]dim F for all( F ∈ S( X )) such that dim F < d. If dim T = d, it is shown that
the boundary δ ψT (B(T )) is a regular CW complex with face poset isomorphic
to S( X )<T (all X-forests in S( X ) less than T ). The poset [0̂, T ] is obtained by
adding a 0̂ and a 1̂ = T . In [22] it was shown that [0̂, T ] is graded
( and thin.
)
If [0̂, T ] also has a recursive coatom ordering, it follows that ψT δ(B(T )) is
(
)
(
)
homeomorphic to δ [0, 1]d . That ψT B(T ) is homeomorphic to [0, 1]d now
(
)
follows from the fact that for each y ∈ ψT B(T ) , ψT−1 (y) is a contractible regular
cell complex (which is shown in [22]).
The main ingredient of the proof is to conclude that all intervals [0̂, F ], where
F ∈ S( X ), have recursive coatom orderings. The method to show this, is not to
find one coatom ordering that is valid in each step of the recursion, but to find
a set of coatom orderings such that in each step of the recursion it is possible to
choose one valid coatom ordering from the set.
This paper is a joint work with Svante Linusson, Vincent Moulton and Mike
Steel, and my main contribution is to prove that each interval [0̂, F ] has a recursive coatom ordering. It is published in Advances in Applied Mathematics,
volume 41 (2008), pages 158–176.
Some rather straightforward proofs, similar to other proofs in the article,
were omitted in the journal version of the paper. For completeness, they are
included here in a supplement in the end of the article, and footnotes are added
at the references to the full proofs.
Paper 3: A generating function for X -forests
The third paper studies a generating function for the elements of the Tuffley
poset. The elements are semi-labelled forests, called X-forests, where X is a
finite set of labels.
Since the edge-product space E ( X ) has a regular cell decomposition with
face poset given by the Tuffley poset S( X ), it is natural to be interested in the
poset itself.
Interesting questions concern, e.g., the number of X-trees or X-forests with
a fixed number of edges, or the number of trees in the X-forests. Also, statistics
like the average number of edges in all X-forests or X-trees with | X | = n, or the
distribution of the number of trees in X-forests, could be interesting to study.
To be able to answer such questions, a useful tool is to have a generating
function which counts the number of edges, labelled nodes, components, and
labels in X.
It happens to be that it is easier to study a generating function counting
unlabelled nodes instead of labelled, but it gives as much information, as the
number of labelled nodes is the same as the number of edges plus the number
of components minus the number of unlabelled nodes.
The generating function studied looks like this:
19
8 Overview of the papers
∞
1+
∑
(
∑
n=1 F ∈S([n])
x |C(F )| y|E(F )| z|V (F )|
) wn
n!
,
where x, y, and z count the number of components, edges, and unlabelled nodes,
respectively, in the [n]-forests.
A closed formula for this generating function is found. It is
(
yz + 1
exp x
2y
(
( ( y(ew − 1 + z)
)2
yz
1
− yz+1 )
−
W
−
e
+
1
0
yz + 1
(yz + 1)2
))
,
where W0 is the principal branch of the Lambert W function.
In addition, closed formulas are found for the generating function of X-trees,
and for the generating functions of X-trees and X-forests with the restriction of
having at most one label on each node.
Since this generating function is analytic in a neighbourhood of (w, x, y, z) =
(0, 1, 1, 1), it is possible to use the powerful tool of singularity analysis to analyse
its coefficients as n → ∞.
Singularity analysis of generating functions is a method that can be used to
compute asymptotic expressions of the coefficients of a generating function by
analysing the singularities of the function. This method is described in [15].
In that way the asymptotic mean, variance, etc. are calculated for the number of edges, components, and unlabelled nodes in X-trees and X-forests as
| X | → ∞.
The asymptotic distributions of edges and unlabelled nodes seem to be normal distributions, while the asymptotic distribution of the number of components amazes with nice rational values on asymptotic mean, variance etc..
Paper 4: Pattern containment in random permutations
In this paper permutation statistics counting occurrences of patterns are studied
on a certain kind of random permutations. Let s be a permutation statistic.
Consider the product π of t permutations chosen uniformly at random from a
subset Γ of the symmetric group Sn . Now s(π ) is a random variable, and an
important characteristic is of course the expected value of s(π ), which will be
denoted EΓ (s, t).
This is interesting for example in phylogenetics, where Γ is mostly taken to
be some set of transpositions. See for example [12], [13], [14], and [19].
There is a method, developed by Hultman in [18], that makes it easy to compute EΓ (s, t) when Γ is a union of conjugacy classes of Sn . The only prerequisite
for using the method is that the mean s of s over the conjugacy classes has to be
expressed as a linear combination of irreducible characters of Sn .
This paper is focused on expressing the means of statistics counting occurrences of classical and vincular patterns as linear combinations of the irreducible
characters.
20
Introduction
A procedure for calculating these expressions when the patterns have length
3 is developed, and is then used to write the means of all statistics counting
occurrences of classical and vincular patterns of length 3 as linear combinations
of irreducible characters. It turns out that only five irreducible characters are
needed in all these expressions.
It is exemplified how the expressions of the means can be used to find the
expected values EΓ (s, t), where Γ is the set of all transpositions in Sn , for all
statistics s counting occurrences of classical patterns of length 3.
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References
21
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Included Papers
The articles associated with this thesis have been removed for copyright
reasons. For more details about these see:
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-98263