Fibonacci numbers, alternating parity sequences and faces of the tridiagonal Birkhoff polytope

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.http://www.sciencedirect.com/science/article/B6V00-4NF4F6K-H/1/e5d0725d5317b08a025d7df94b2ca64

Computational aspects of lattice theory

The use of computers to produce a user-friendly safe environment is an important area of research in computer science. This dissertation investigates how computers can be used to create an interactive environment for lattice theory. The dissertation is divided into three parts. Chapters two and three discuss mathematical aspects of lattice theory, chapter four describes methods of representing and displaying distributive lattices and chapters five, six and seven describe a definitive based environment for lattice theory. Chapter two investigates lattice congruences and pre-orders and demonstrates that any lattice congruence or pre-order can be determined by sets of join-irreducibles. By this correspondence it is shown that lattice operations in a quotient lattice can be calculated by set operations on the join-irreducibles that determine the congruence. This alternative characterisation is used in chapter three to obtain closed forms for all replacements of the form "h can replace g when computing an element f", and hence extends the results of Beynon and Dunne into general lattices. Chapter four investigates methods of representing and displaying distributive lattices. Techniques for generating Hasse diagrams of distributive lattices are discussed and two methods for performing calculations on free distributive lattices and their respective advantages are given. Chapters five and six compare procedural and functional based notations with computer environments based on definitive notations for creating an interactive environment for studying set theory. Chapter seven introduces a definitive based language called Pecan for creating an interactive environment for lattice theory. The results of chapters two and three are applied so that quotients, congruences and homomorphic images of lattices can be calculated efficiently

A tight Monte-Carlo algorithm for Steiner Tree parameterized by clique-width

Recently, Hegerfeld and Kratsch [ESA 2023] obtained the first tight algorithmic results for hard connectivity problems parameterized by clique-width. Concretely, they gave one-sided error Monte-Carlo algorithms that given a $k$-clique-expression solve Connected Vertex Cover in time $6^kn^{O(1)}$ and Connected Dominating Set in time $5^kn^{O(1)}$. Moreover, under the Strong Exponential-Time Hypothesis (SETH) these results were showed to be tight. However, they leave open several important benchmark problems, whose complexity relative to treewidth had been settled by Cygan et al. [SODA 2011 & TALG 2018]. Among which is the Steiner Tree problem. As a key obstruction they point out the exponential gap between the rank of certain compatibility matrices, which is often used for algorithms, and the largest triangular submatrix therein, which is essential for current lower bound methods. Concretely, for Steiner Tree the $GF(2)$-rank is $4^k$, while no triangular submatrix larger than $3^k$ was known. This yields time $4^kn^{O(1)}$, while the obtainable impossibility of time $(3-\varepsilon)^kn^{O(1)}$ under SETH was already known relative to pathwidth. We close this gap by showing that Steiner Tree can be solved in time $3^kn^{O(1)}$ given a $k$-clique-expression. Hence, for all parameters between cutwidth and clique-width it has the same tight complexity. We first show that there is a ``representative submatrix'' of GF(2)-rank $3^k$ (ruling out larger triangular submatrices). At first glance, this only allows to count (modulo 2) the number of representations of valid solutions, but not the number of solutions (even if a unique solution exists). We show how to overcome this problem by isolating a unique representative of a unique solution, if one exists. We believe that our approach will be instrumental for settling further open problems in this research program

Computational Group Theory (hybrid meeting)

This was the eighth Oberwolfach Workshop on Computational Group Theory. It demonstrated how an increasing number and variety of deep theoretical results are being used to devise powerful and practical algorithms in Computational Group Theory. The talks also presented connections with and applications to Number Theory, Combinatorics, Geometry, and Geometric Group Theory

Classification of Distinct Fuzzy Subgroups of the Dihedral Group Dp nq for p and q distinct primes and n ∈ N

In this dissertation, we classify distinct fuzzy subgroups of the dihedral group Dpnq, for p and q distinct primes and n ∈ N, under a natural equivalence relation of fuzzy subgroups and a fuzzy isomorphism. We aim to present formulae for the number of maximal chains and the number of distinct fuzzy subgroups of this group. Our study will include some theory on non-abelian groups since the classification of distinct fuzzy subgroups of this group relies on the crisp characterization of maximal chains. We give the definition of a natural equivalence relation introduced by Murali and Makamba in [67] which we will use in this study. Based on this definition, we introduce two counting techniques that we will use to compute the number of distinct fuzzy subgroups of Dpnq. In this dissertation, we use the criss-cut counting technique as our primary method of enumeration, and the cross-cut method serves as a means of verifying results we obtain from our primary method. To classify distinct fuzzy subgroups of this group, we begin by investigating the dihedral groups Dpnq, for p and q distinct primes and specific values of n = 2 and 3 to observe a trend. We classify the flags of these groups using the characterization of flags introduced in [93]. From this characterization, we then present formulae for the number of distinct fuzzy subgroups attributed to the flags of Dp 2q and Dp 3q . To generalise results for Dpnq, for p and q distinct primes and n ∈ N, we characterize the flags of this group and classify them as either cyclic, mdcyclic for 1 ≤ m ≤ n, or b-cyclic. Finally, we establish a general formula for the number of distinct fuzzy subgroups obtainable from these flags. We conclude by comparing results obtained from using our general formula to those obtained by other researchers for the same group. Based on the results from this study, we give an outline of future research wor