Enumerative combinatorics, continued fractions and total positivity

Determining whether a given number is positive is a fundamental question in mathematics. This can sometimes be answered by showing that the number counts some collection of objects, and hence, must be positive. The work done in this dissertation is in the field of enumerative combinatorics, the branch of mathematics that deals with exact counting. We will consider several problems at the interface between enumerative combinatorics, continued fractions and total positivity. In our first contribution, we exhibit a lower-triangular matrix of polynomials in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. This generalises Brenti’s conjecture from 1996. We prove the coefficientwise total positivity of a three-variable case which includes the reversed Stirling subset triangle. Our next contribution is the study of two sequences whose Stieltjes-type continued fraction coefficients grow quadratically; we study the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations, a class of permutations of 2n, with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings. After this, we interpret the Foata–Zeilberger bijection in terms of Laguerre digraphs, which enables us to count cycles in permutations. Using this interpretation, we obtain Jacobi-type continued fractions for multivariate polynomials enumerating permutations, and also Thron-type and Stieltjes-type continued fractions for multivariate polynomials enumerating D-permutations, in both cases including the counting of cycles. This enables us to prove some conjectured continued fractions due to Sokal–Zeng from 2022, and Randrianarivony–Zeng from 1996. Finally, we introduce the higher-order Stirling cycle and subset numbers; these generalise the Stirling cycle and subset numbers, respectively. We introduce some conjectures which involve different total-positivity questions for these triangular arrays and then answer some of them

End-to-end distance vector distribution with fixed end orientations for the wormlike chain model

We find exact expressions for the end-to-end distance vector distribution function with fixed end orientations for the wormlike chain model. This function in Fourier-Laplace space adopts the form of infinite continued fractions, which emerges upon exploiting the hierarchical structure of the moment-based expansion. Our results are used to calculate the root-mean-square end displacement in a given direction for a chain with both end orientations fixed. We find that the crossover from rigid to flexible chains is marked by the root-mean-square end displacement slowly losing its angular dependence as the coupling between chain conformation and end orientation wanes. However, the coupling remains strong even for relatively flexible chains, suggesting that the end orientation strongly influences chain conformation for chains that are several persistence lengths long. We then show the behavior of the distribution function by a density plot of the probability as a function of the end-to-end distance vector for a wormlike chain in two dimensions with one end pointed in a fixed direction and the other end free (in its orientation). As we progress from high to low rigidity, the distribution function shifts from being peaked at a location near the full contour length of the chain in the forward direction, corresponding to a straight configuration, to being peaked near zero end separation, as in the Gaussian limit. The function exhibits double peaks in the crossover between these limiting behaviors

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics