The k-assignment polytope, phylogenetic trees, and permutation patterns

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 ⊆ S n , where Γ is a union of conjugacy classes, are considered. Hultman has described a method for computing such an expected value, denoted E Γ (s, t), of a statistic s, when Γ is a union of conjugacy classes of S n . The only prerequisite is that the mean of s over the conjugacy classes is written as a linear combination of irreducible characters of S n . 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 avstån-det mellan tv...

Counting Phylogenetic Networks

“…For example, Bona and Flajolet [3] find the asymptotic probability that two random trees fromT are isomorphic when we remove the labels on the leaves, and Gill [6] estimates parameters of X-trees and X-forests (which are related to phylogenetic trees with leaf-set X).…”

Positivity of permutation pattern character polynomials

“…More recently, efforts have been made to understand the distribution of pattern occurrences on conjugacy classes in S n using the character theory of the symmetric group. Hultman [5] and Gill [4] considered the mean of N σ on conjugacy classes for the cases k = 2 and k = 3, respectively. The first author and Ryba [3] used a new approach involving partition algebras to prove that all moments of N σ on conjugacy classes are polynomials in n, m 1 , .…”

“…so we have F (x, y) = 1 2x 2 (1 + y 2 ) + x4 (1 − y 2 ) 2 and thusx 2 y 2 F (x, y) 2 = x 2 y 2 1 − 2x 2 (1 + y 2 ) + x 4 (1 − y 2 ) 2x 2 − 2x 2 y 2 + x 4 y 4 − 2x 4 y 2 + x x 2 − x 2 y 2 ) 2 − (2x 2 y) 2 = G(x, y),which completes the proof. Now for our third part of our proof, we will define an additional function E ′ (n, k, r) and simultaneously prove formulas for both E(n, k, r) and E ′ (n, k, r) by induction on n.Definition A.3.…”

Positivity of permutation pattern character polynomials

“…Several authors have initiated the study of such interactions by investigating the relationships between pattern occurrences and conjugacy classes in the symmetric group. For example, in 2013-2014, Hultman and Gill investigated the expected value of N σ (π) for π restricted to certain conjugacy classes in the case where σ ∈ S 2 or S 3 [4]. In 2021, Gaetz and Ryba showed that the dth moments of N σ on conjugacy classes in S n are given by polynomials in n, m 1 , .…”

“…where the sum is taken over partitions µ ⊢ l − 1 whose young diagrams are contained in that of λ. 4 λ âλ…”