On two unimodal descent polynomials

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.Comment: 16 pages, 4 figure

On the number of rectangulations of a planar point set

AbstractWe investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n+1)st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schröder number, and the total number of rectangulations is O(20n/n4)

The Brownian limit of separable permutations

We study random uniform permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a "Brownian separable permuton".Comment: 45 pages, 14 figures, incorporating referee's suggestion

Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration

We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior. This strange behavior even provides some very unexpected data related to the number of 1324-avoiding permutations

Separable elements and splittings in Weyl groups of Type BB

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair $(X,Y)$ of subsets of the symmetric group $\mathfrak{S}_n$, the multiplication map $X\times Y\rightarrow \mathfrak{S}_n$ is a splitting (i.e., a length-additive bijection) of $\mathfrak{S}_n$ if and only if $X$ is the generalized quotient of $Y$ and $Y$ is a principal lower order ideal in the right weak order generated by a separable element. They conjectured this result can be extended to all finite Weyl groups. In this paper, we classify all separable and minimal non-separable signed permutations in terms of forbidden patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type $B$.Comment: 20 pages, 2 figures, comments welcom

Extremal and probabilistic bootstrap percolation

In this dissertation we consider several extremal and probabilistic problems in bootstrap percolation on various families of graphs, including grids, hypercubes and trees. Bootstrap percolation is one of the simplest cellular automata. The most widely studied model is the so-called r-neighbour bootstrap percolation, in which we consider the spread of infection on a graph G according to the following deterministic rule: infected vertices of G remain infected forever and in successive rounds healthy vertices with at least r already infected neighbours become infected. Percolation is said to occur if eventually every vertex is infected. In Chapter 1 we consider a particular extremal problem in 2-neighbour bootstrap percolation on the n \times n square grid. We show that the maximum time an infection process started from an initially infected set of size n can take to infect the entire vertex set is equal to the integer nearest to (5n^2-2n)/8. In Chapter 2 we relax the condition on the size of the initially infected sets and show that the maximum time for sets of arbitrary size is 13n^2/18+O(n). In Chapter 3 we consider a similar problem, namely the maximum percolation time for 2-neighbour bootstrap percolation on the hypercube. We give an exact answer to this question showing that this time is \lfloor n^2/3 \rfloor. In Chapter 4 we consider the following probabilistic problem in bootstrap percolation: let T be an infinite tree with branching number \br(T) = b. Initially, infect every vertex of T independently with probability p > 0. Given r, define the critical probability, p_c(T,r), to be the value of p at which percolation becomes likely to occur. Answering a problem posed by Balogh, Peres and Pete, we show that if b \geq r then the value of b itself does not yield any non-trivial lower bound on p_c(T,r). In other words, for any \varepsilon > 0 there exists a tree T with branching number \br(T) = b and critical probability p_c(T,r) < \varepsilon. However, in Chapter 5 we prove that this is false if we limit ourselves to the well-studied family of Galton--Watson trees. We show that for every r \geq 2 there exists a constant c_r>0 such that if T is a Galton--Watson tree with branching number \br(T) = b \geq r then $$p_c(T,r) > \frac{c_r}{b} e^{-\frac{b}{r-1}}.$$ We also show that this bound is sharp up to a factor of O(b) by describing an explicit family of Galton--Watson trees with critical probability bounded from above by C_r e^{-\frac{b}{r-1}} for some constant C_r>0

Applying the Free-Energy Principle to Complex Adaptive Systems

The free energy principle is a mathematical theory of the behaviour of self-organising systems that originally gained prominence as a unified model of the brain. Since then, the theory has been applied to a plethora of biological phenomena, extending from single-celled and multicellular organisms through to niche construction and human culture, and even the emergence of life itself. The free energy principle tells us that perception and action operate synergistically to minimize an organism’s exposure to surprising biological states, which are more likely to lead to decay. A key corollary of this hypothesis is active inference—the idea that all behavior involves the selective sampling of sensory data so that we experience what we expect to (in order to avoid surprises). Simply put, we act upon the world to fulfill our expectations. It is now widely recognized that the implications of the free energy principle for our understanding of the human mind and behavior are far-reaching and profound. To date, however, its capacity to extend beyond our brain—to more generally explain living and other complex adaptive systems—has only just begun to be explored. The aim of this collection is to showcase the breadth of the free energy principle as a unified theory of complex adaptive systems—conscious, social, living, or not

Molecular Dynamics Simulation

Condensed matter systems, ranging from simple fluids and solids to complex multicomponent materials and even biological matter, are governed by well understood laws of physics, within the formal theoretical framework of quantum theory and statistical mechanics. On the relevant scales of length and time, the appropriate ‘first-principles’ description needs only the Schroedinger equation together with Gibbs averaging over the relevant statistical ensemble. However, this program cannot be carried out straightforwardly—dealing with electron correlations is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the many-body problem can be effectively reduced to one of independent particles or quasi-particles. [...