The k -assignmentPolytope,PhylogeneticTrees,andPermutationPatterns

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

Thek-assignement Polytope, Phylogenetic Trees, and Permutation Patterns Jonna Gill

Matematiska institutionen Linköpings universitet SE-581 83 Linköping, Sweden jonna.gill@liu.se

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 ver-tices 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 repre-sentation of the faces by certain bipartite graphs is given. This reprerepre-sentation 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 edge-product 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 stud-ied. 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, denotedE_Γ(s, t), of a statistic s, whenΓ 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 combina-tion 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.

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 Birk-hoffpolytopen. 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 avstån-det mellan två hörn, där avstånavstån-det mellan två hörn är avstån-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 si-dor (kanterna), sex 2-dimensionella sisi-dor (det vi vanligtvis kallar sisi-dor), och en 3-dimensionell sida (hela mjölkpaketet). Diametern av ett mjölkpaket är 3.

Birkhoffpolytopen är polytopen vars hörns koordinater i n2dimensioner 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 koatomord-ning, och att detta medför mycket trevliga egenskaper hos den matematiska mo-dellen för mutationer.

En genererande funktion för X-skogar (som räknar antalet X-skogar med av-seende 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

vi Populärvetenskaplig sammanfattning

talen i någon ordning. En permutationπ av{1, 2, . . . , n}kan skrivasπ1π2. . .πn.

Ett permutationsmönster är en sekvens p1p2. . . pk, där k är längden på mönstret,

och mönstret sägs förekomma i permutationen π om det finns en delsekvens πi1πi2. . .πik iπ med samma relativa ordning som p1p2. . . 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 förekoms-ter av ett visst permutationsmönsförekoms-ter 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 delsekven-sen ska börja med π1) beskrivs, och alla deras medelvärdesfunktioner skrivs i

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 tak-ing over the supervision when Svante Linusson left Linköptak-ing University for KTH Royal Institute of Technology. I would also like to thank my present sec-ond 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.

Contents

I

Introduction

1

4.1 Basic theory of polytopes . . . 7

4.2 Transportation polytopes and network flow polytopes . . . 8

4.3 The Birkhoff polytope . . . 9

5 The edge-product space and its face poset . . . 10

5.1 Finite CW complexes . . . 10

5.2 The edge-product space of phylogenetic trees . . . 11

5.3 The Tuffley poset . . . 12

6 The Lambert W function . . . 13

7 Permutations . . . 13

7.1 The group Sn of permutations . . . 13

7.2 Representations of Sn and their characters . . . 14

7.3 Permutation statistics . . . 15

8 Overview of the papers . . . 17

Paper 1: The k-assignment polytope . . . . 17

Paper 2: A regular decomposition of the edge-product space of phylogenetic trees . . . 17

Paper 3: A generating function for X-forests . . . . 18

Paper 4: Pattern containment in random permutations . . . 19

References . . . 20

x Contents

II

Papers

23

1 Thek-assignment polytope 25

1 Introduction . . . 27

2 Some basic properties of the k-assignment polytope . . . . 28

3 Description of the face poset . . . 30

4 The diameter of M(m, n, k) . . . 36

5 Ear decomposition . . . 43

Appendix: Proof of Lemma 5.8 . . . 45

References . . . 48

2 A regular decomposition of the edge-product space of phylogenetic trees 51 1 Introduction . . . 54

2 Trees, forests and the Tuffley poset . . . 55

3 Recursive coatom orderings . . . 57

3.1 Preliminaries and definitions . . . 58

3.2 Outline of proof of Theorem 3.1 . . . 60

3.3 There is a recursive coatom ordering for[ˆ0,Γ] . . . 60

3.4 An example of a coatom ordering satisfying(V3) . . . 62

4 The edge-product space is a regular cell complex . . . 62

5 Proof of some combinatorial lemmas . . . 65

5.1 Reformulation of(V1)with implications . . . 66

5.2 Common elements of[ˆ0,α1]and[ˆ0,α2] . . . 66

5.3 AandB are compatible with(V3). . . 67

5.4 The order of the coatoms near two vertices . . . 71

Appendix . . . 71

Supplement . . . 73

References . . . 77

3 A generating function forX-forests 79 1 Introduction . . . 81

2 A generating function for X-forests . . . . 83

3 Recurrence relations for the coefficients . . . 85

4 The generating functions . . . 87

5 Statistics . . . 89

5.1 Statistics of generating functions . . . 89

5.2 Preliminaries on singularity analysis . . . 90

5.3 Singular expansions of derivatives of F and T . . . 93

5.4 Asymptotic statistics of the generating functions . . . 95

Appendix . . . 98

A.1 Computation of the constants pj in Lemma 5.11 . . . 98

Contents xi

References . . . 101

4 Pattern containment in random permutations 103

1 Introduction . . . 105 2 Preliminaries . . . 107 3 Mean statistics of 3-patterns . . . 108 4 A procedure for calculating the means of vincular 3-patterns . . . 114 5 Expected values . . . 120 Appendix . . . 122 References . . . 126

Part I

Introduction

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 x̸= 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 ele-ments 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

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 ˆ0 ∈ S and a unique maximal element ˆ1∈S such that ˆ0≤x≤ ˆ1 for all x∈S. It is possible to add an artificial ˆ0 and/or ˆ1 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 ˆ0 and a ˆ1. 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 ˆ0 and a ˆ1. If S is a lattice, then the minimal elements of S∖ˆ0 are called atoms, and the maximal elements of S∖ˆ1 are called coatoms.

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, . . . ,αtof its coatoms that satisfies the following two conditions:

(i) For all i< j andγ<α_i,αj there is a k<j and an element β such that β is

covered byαk andαj andγ≤β.

(ii) For all j=1, . . . , t, the interval[ˆ0,αj]admits a recursive coatom ordering in

which the coatoms that come first in the ordering are those that are covered by someαkwhere 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 denotedδ(G)and is defined as the smallest numberδ such that any two nodes in G can be connected by a path with at mostδ 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 nec-essary 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 nota-tion is used: If E1and E2are subsets of E(G), then E1+E2denotes 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 G_s−1= x+E1+· · · +Es−1has 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 G_s−1. The decomposition Gs =x+E1+· · · +Eswill 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.

4 Polytopes 7

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⊆Rdthat is the convex hull of a finite point set,

P=Conv(V1, . . . , Vn) for some V1, . . . , Vn∈Rd

or, equivalently, a subset P⊆Rdthat 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 y̸=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≤c0for 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 ⊆Rdbe 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 2 4 15 24 123 234 Ø P 1245 135 ₃₄₅ 45 35 34 23 13 12 3 5 1 1 ₂ 3 4 5 P 3 2 4 1 5

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 denotedδ(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 ware-house produces ri >0 units of the product per time unit, and the j:th customer

requires aj >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

transporta-tion cost. Hence the transportatransporta-tion problem of order m×n is to minimise the linear function m

∑

∑

subject to the conditions

n

∑

∑

The set of matrices(xij)m×nsatisfying these conditions is called the transportation

4 Polytopes 9

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 xijdenotes the flow on the edge (i, j)∈ A. The set of possible solutions to

∑

∑

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 poly-tope. 3 b3 b4 5 b5 1 2 b1 b2 (0, )8 (0, )8 (0, )8 (0, )8 (0, )8 (0, )8 1 2 3 r1 r2 r₃ 1 2 12 l ,u ( ) 13 35 l ,u 43 45 4 24 l ,u ( ) l ,u ( )l ,u34 ( )l ,u₁₃ ( )₃₅ ( )45 l ,u ( )43 34 24 12 1 2 −a −a

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 poly-topes. The m warehouses are represented by m supply nodes with bi=riand the

n customers are represented by n demand nodes with bj = −aj. 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 Sdare described in Section 7.1.)

10 Introduction Definition 4.8. For every permutationσ∈Sd, construct a d×d matrix Xσ by

X_ijσ= {

1 ifσ(i) =j 0 otherwise.

The matrices Xσare 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 inRd×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{Xσ :σ∈Sd} ⊆Rd×d.

The Birkhoff polytope Bd has d! vertices, d2 facets, and dimension(d−1)2.

Two vertices Xσand Xπare 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(B}

d) is 1 if

d≤3, and 2 if d≥4. The points in Bdare precisely

{ X∈Rd×d : xij ≥0 for all i, j, d

∑

∑

Hence the Birkhoff polytope Bdis the transportation polytope T(r, a)where

m=n=d, all ri=1, and all aj=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 FB(G), then the vertices of FB(G)are exactly the vertices that correspond to all perfect matchings P such that P⊆ G. In that way the face poset of Bdis isomorphic to the lattice of all elementary subgraphs

of Kd,dordered by inclusion.

Remember that all components of elementary graphs have ear decomposi-tions. The following relationship between the dimension of a face in Bdand 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 FB(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].

5 The edge-product space and its face poset 11

The closed d-ball Bdis defined to be the set{x ∈Rd :|x| ≤1}(where|·|is the standard norm inRd).

A topological space Y is a Hausdorff space if, for each x, y∈Y such that x̸=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 subsetσ is called an open d-cell if there exists a continuous mappingψ : 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 closureσ is the corresponding closed cell. In fact, σ= ψ(Bd). Let δ(σ)denote the boundaryσ∖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(σ_α)⊆ C<dimσα_{(the union of all cells in}C_{of dimension less than dimσα) 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 mappingψα: Bdimσα_{→Y can be chosen to be a homeomorphism on}

all of Bdimσα_{, thenC 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. Letλ be a map from E(T)to[0, 1]. Define a new map p(T,λ)from

(_X 2 ) to[0, 1]by p(T,λ)(x, y) =

∏

where P(T; x, y)is the set of edges in the path in T from x to y. LetE(X, T)⊂ [0, 1](X2)denote the image of the map

ΛT:[0, 1]E(T) → [0, 1](

X

2)_, λ7→ _p(T,λ)

and letE(X)denote the union of the subspacesE(X, T)of[0, 1](X2)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 ofE(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-treeT is a pair(T;ϕ), where T is a tree andϕ : 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 collectionF = {(A,TA): A ∈ µ}where µ is a

set partition of X (a collection of disjoint subsets of X whose union is X) andTA

is an A-tree for each A∈µ.

1 5 7 1,3,8,11 4 10 6 9 2 3 2, 9 4,7,10 5 8 6 11

Figure 7:An X-tree and an X-forest, where X={1, 2, . . . , 11}.

To remove an edge e= (u, v)from an X-forestT = (T;ϕ)and identify u and v, labelling the new vertexϕ−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 byF2≤ F1ifF2can be obtained fromF1by 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,TA) : A ∈ µ}, define

B(F) = ∏_A∈µB(TA), so that Int

(

B(F)) = ∏_A∈µInt(B(TA)

)

. The sets B(F) and Int(B(F))are homeomorphic to a closed ball and an open ball, respectively, of dimension∑A∈µ|E(TA)|(this quantity will be called the dimension ofF).

Given an X-tree T = (T;ϕ) and a map λ : E(T) → [0, 1], define the map p(T ,λ): (_X 2 ) → [0, 1]by p(T ,λ)(x, y) =

∏

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. Ifϕ(x) =ϕ(y), then p(T ,λ)(x, y):=1.

6 The Lambert W function 13

For an X-forest F = {(A,TA) : A ∈ µ}and a map λ = (λA, A ∈ µ), let

ψF : B(F) → [0, 1](X2)be defined by

ψF(λ)(x, y) = {

p_(TA,λA)(x, y) if∃A∈µ such that x, y∈ A,

0 otherwise.

The edge-product spaceE(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 ζ̸=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]−∞,−1_e], it is defined at

−1

e 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 function is T(z) =_∑∞_n=1tnz

n

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

S

of permutations

Definition 7.1. A permutation of the set[n] := {1, 2, . . . , n}is a linear ordering π1π2. . .πn of the elements of[n]. The expression π1π2. . .πn is called a word,

and the elementsπi are consequently called letters. The permutationπ can also

be seen as a bijective functionπ :[n]→ [n]given byπ(i) =π_i.

The inverse π−1 of a permutation π = π1π2. . .πn is given byπ−1(πi) = i,

and the product of two permutationsπ 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(i1i2i3 . . . ik)is a cycle

ofπ, 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(i2i3 . . . ik i1). It is easy to

see that every element of[n]appears in a unique cycle ofπ.

As an example, ifπ ∈ S7is 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=1λj = n andλ1≥ λ2≥ · · · ≥ λk. Thatλ is a

partition of n is denotedλ⊢n.

The sequence of the lengths of all cycles inπ in weakly decreasing order is called the cycle type ofπ. 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 ofσπσ−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 SntoC that is constant

on conjugacy classes. A class function can also be seen as a function from the partitions of n toC.

7.2

Representations of

S

and their characters

Now the representations of Snand their characters will be briefly described. For

the full details, see [23]. Let GLddenote 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), andρ(πτ) = ρ(π)ρ(τ) for allπ, τ ∈ Sn. The

character of the representation ρ is the map χ : Sn → C which is defined by

χ(π) =trρ(π), where tr denotes the trace of a matrix.

Two matrix representationsρ1andρ2of Snare equivalent if and only if there

exists a fixed matrix T such thatρ2(π) =Tρ1(π)T−1for allπ∈Sn. In that case

their characters are identical.

It is easy to see that all characters of representations of Snare class functions,

sinceχ(σπσ−1) =trρ(σπσ−1) =trρ(σ)ρ(π)ρ(σ)−1=trρ(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.

7 Permutations 15

There are as many irreducible representations of Sn as there are conjugacy

classes, and there is a standard bijection between them. The irreducible repre-sentation corresponding to the conjugacy class with cycle typeλ will be denoted ρλ_{, and the character of that representation will be denotedχ}λ_.

Definition 7.5. Let f and g be any two functions from Sn toC. The inner product

of f and g is

⟨f , g⟩ = 1 n!_π∈S

∑

n

f(π)g(π)∗, where g(π)∗is the complex conjugate of g(π).

Theorem 7.6. The irreducible characters of Snform 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 C_λ is the conjugacy class with cycle typeλ, then

s(λ) = 1

|C_λ|_π∈C

∑

λ

s(π).

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=π1π2. . .πnis a permutation in Sn:

• The index i is a descent ofπ if πi >πi+1. The permutation statistic counting

the number of descents ofπ is often denoted des(π). The mean can be written as des= n−1₂ χ(n)₋1

nχ(n−1,1)−n1χ(n−2,1,1)(see [16]).

• 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 asc= n−1₂ χ(n)+1_nχ(n−1,1)+_n1χ(n−2,1,1).

• The pair(πi,πj)is an inversion ofπ if i< j andπi >πj. The mean of the

number of inversions is inv = n(n−1)₄ χ(n)− n+1₆ χ(n−1,1)−1₆χ(n−2,1,1) (see [18]).

16 Introduction

• A fixed point ofπ is an index i such that πi =i. The number of fixed points

ofπ is the number of 1-cycles, so this is a class function and can be written asχ(n)+χ(n−1,1) (see [19]).

Definition 7.9. A permutationπ that swaps two elements is called a transposi-tion. This means thatπ(i) = i for all i ∈ [n]except two, call them i1 and i2, for

whichπ(i1) = i2 and π(i2) = i1. If i2 and i1 are adjacent, thenπ 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 = ϕ₁ϕ₂. . .ϕk ∈ Sk in a

permutation π = π₁π₂. . .πn ∈ Sn is a subsequence in π of length k whose

letters are in the same relative order as those inϕ.

For example, an occurrence of the classical pattern 123 inπ∈Sn is a

subse-quenceπi1πi2πi3, where i1<i2<i3, such thatπi1 <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 patternϕ is written as a permutation in Skenclosed 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 ofϕ in π must end with πn.

As an example, an occurrence of the vincular pattern [1-23)in π ∈ Sn is a

subsequence πi1πi2πi3, where i1 < i2 < i3, i1 = 1, and i3 = i2+1, such that

πi1 <πi2 <πi3.

Let the number of occurrences of a patternϕ in π be denoted pat_ϕ(π). Many permutation statistics can be written as sums of statistics counting occurrences of vincular patterns. Consider for example the following statistic: A letterπiin

a permutationπ is a peak if πi−1<πi>πi+1. If peak(π)is the number of peaks

inπ, 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 permutationsπ ∈ Sn with inv(π) =k is equal to the

number of permutationsτ∈Snwith 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 equiva-lences) 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 optimisa-tion 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 representa-tion as doped elementary graphs is used to describe the cover relarepresenta-tion 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 spaceE(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 thatE(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 ψ_F(B(F)) is homeomorphic to [0, 1]dimF 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 dimT is used, with the induction hypothesis that the setψF(B(F))is homeomorphic to [0, 1]dimF for allF ∈S(X)such that dimF <d. If dimT =d, it is shown that the boundaryδ(ψ_T(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[ˆ0,T ]is obtained by adding a ˆ0 and a ˆ1 = T. In [22] it was shown that [ˆ0,T ] is graded and thin. If [ˆ0,T ] also has a recursive coatom ordering, it follows that ψT(δ(B(T ))) is homeomorphic to δ([0, 1]d)_{. That ψ}

T(B(T )) is homeomorphic to [0, 1]d _now

follows from the fact that for each y∈ψT(B(T )),ψ−1_T (y)is a contractible regular cell complex (which is shown in [22]).

The main ingredient of the proof is to conclude that all intervals[ˆ0,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 [ˆ0,F] has a re-cursive 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.

8 Overview of the papers 19 1+ ∞

∑

∑

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

exp ( xyz+1 2y ( 1 (yz+1)2− ( W0 ( −y(ew−1+z) yz+1 e − yz yz+1)₊₁ )2)) , where W0is 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 num-ber 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 nor-mal distributions, while the asymptotic distribution of the number of compo-nents 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 denotedE_Γ(s, t).

This is interesting for example in phylogenetics, whereΓ 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 com-puteE_Γ(s, t)whenΓ 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 occur-rences 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 E_Γ(s, t), where Γ is the set of all transpositions in Sn, for all

statistics s counting occurrences of classical patterns of length 3.

References

[1] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network flows: theory, algorithms, and applications, Prentice Hall, Upper Saddle River, NJ, 1993.

[2] E. Babson and E. Steingrímsson, Generalized permutation patterns and a classi-fication of the Mahonian statistics, Sém. Lothar. Combin. 44 (2000), Art. B44b. [3] M. Bóna, Combinatorics of permutations, Discrete Mathematics and its

Appli-cations 72, CRC Press, Boca Raton, FL, 2012.

[4] L. J. Billera and A. Sarangarajan, The combinatorics of permutation polytopes, Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994) DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 24, Amer. Math. Soc., Providence, RI, 1996, 1–23.

[5] A. Björner, Topological methods, Handbook of Combinatorics, Vol. 2, MIT Press, Cambridge, MA, 1995, pp. 1819–1872.

[6] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler, Oriented matroids, Cambridge University Press, Cambridge, 1993.

[7] R. A. Brualdi and P. M. Gibson, Convex polyhedra of doubly stochastic matrices, I. Applications of the permanent function, J. Combin. Theory, Ser. A 22 (1977), no. 2, 194–230.

[8] R. A. Brualdi and P. M. Gibson, Convex polyhedra of doubly stochastic matrices, II. Graph ofΩn, J. Combin. Theory, Ser. B 22 (1977), no. 2, 175–198.

[9] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996), no. 4, 329–359. [10] R. M. Corless, D. J. Jeffrey, and D. E. Knuth, A sequence of series for the

Lambert W function, International symposium on symbolic and algebraic computation (Kihei, HI, 1997), ACM, New York, 1997, pp. 197–204.

[11] V. A. Emeliˇcev, M. M. Kovalev, and M. K. Kravtsov, Polytopes, graphs and optimisation, Cambridge University Press, Cambridge, 1984.

References 21

[12] N. Eriksen, Expected number of inversions after a sequence of random adja-cent transpositions — an exact expression, Discrete Math. 298 (2005), no. 1–3, 155–168.

[13] N. Eriksen and A. Hultman, Estimating the expected reversal distance after a fixed number of reversals, Adv. in Appl. Math. 32 (2004), no. 3, 439–453. [14] H. Eriksson, K. Eriksson, and J. Sjöstrand, Expected number of inversions after

a sequence of random adjacent transpositions, Formal power series and alge-braic combinatorics (Moscow, 2000), Springer, Berlin, 2000, pp. 677–685. [15] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University

Press, Cambridge, 2009.

[16] J. Fulman, The distribution of descents in fixed conjugacy classes of the symmetric groups, J. Combin. Theory Ser. A 84 (1998), no. 2, 171–180.

[17] J. Gill, The k-assignment polytope and the space of evolutionary trees, Licentiate thesis, Linköping Studies in Science and Technology. Theses No. 1117, Linköping University, 2004. Available from Liu Electronic Press at http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5677

[18] A. Hultman, Permutation statistics of products of random permutations, arXiv:1301.0430 [math.CO], 2013.

[19] A. Jönsson, Evolutionary fixed point distance problems, M.Sc. Thesis, Chalmers University of Technology, Gothenburg, 2009.

[20] D. E. Knuth, The art of computer programming, Vol. 1, 2nd ed., Addison-Wesley, Reading, MA, 1973.

[21] L. Lovász and M. D. Plummer, Matching theory, North-Holland Mathemat-ics Studies 121, Annals of Discrete MathematMathemat-ics 29, North-Holland, Ams-terdam, 1986.

[22] V. Moulton and M. Steel, Peeling phylogenetic ‘oranges’, Adv. in Appl. Math. 33 (2004), no. 4, 710–727.

[23] B. E. Sagan, The symmetric group: representations, combinatorial algorithms, and symmetric functions, 2nd ed., Graduate Texts in Mathematics 203, Springer-Verlag, New York, 2001.

[24] C. Semple and M. Steel, Phylogenetics, Oxford Lecture Series in Mathematics and its Applications 24, Oxford University Press, Oxford, 2003.

[25] R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin. 6 (1985), no. 4, 383–406.

[26] R. P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, Cambridge, 1997. [27] G. M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics 152,

Included Papers

The articles associated with this thesis have been removed for copyright

reasons. For more details about these see: