Enumeration of simple random walks and tridiagonal matrices

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .Comment: several ref.and comments added, misprints correcte

Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers)

In this paper, we describe how to find the generating function for the sum of the areas under Motzkin paths of length $n$ through a method that uses dynamical programming, which we show can be expanded for paths with any given set of steps that start and end at height zero and never have a negative height. We then explore the case where, instead of a set of steps, we are given the quadratic functional equation $f(x,q)=P(x,q)+Q(x,q)f(x,q)+R(x,q)f(x,q)f(qx,x)$. We present a fully automated method for finding perturbation expansions of the solutions $f(x,q)$ to such quadratic functional equations and demonstrate this method using Motzkin paths. More importantly, we combine computer algebra with calculus to automatically find $\frac{d^k}{dq^k}\left[f(x,q) \right]\big|_{q=1}$, explicitly expressed in terms of radicals. We use Dyck and Motzkin paths to exemplify how this can be used to find explicit generating functions for the sum of the areas under such paths and for the sum of a given power of the areas

Exact Partition Function for the Random Walk of an Electrostatic Field

The partition function for the random walk of an electrostatic field produced by several static parallel infinite charged planes in which the charge distribution could be either ±σ is obtained. We find the electrostatic energy of the system and show that it can be analyzed through generalized Dyck paths. The relation between the electrostatic field and generalized Dyck paths allows us to sum overall possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example

Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces

The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(h_m,L) of the maximal height h_m (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L where the function f(x) is the Airy distribution function. This result is valid for both the Edwards-Wilkinson and the Kardar-Parisi-Zhang interfaces. For the free boundary case, the same scaling holds P(h_m,L)=L^{-1/2}F(h_m L^{-1/2}), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the Edwards-Wilkinson interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [ S.N. Majumdar and A. Comtet, Phys. Rev. Lett., 92, 225501 (2004)].Comment: 27 pages, 10 .eps figures included. Two figures improved, new discussion and references adde

Generating functions and the enumeration of lattice paths

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science. Johannesburg, 2013.Our main focus in this research is to compute formulae for the generating function of lattice paths. We will only concentrate on two types of lattice paths, Dyck paths and Motzkin paths. We investigate di erent ways to enumerate these paths according to various parameters. We start o by studying the relationship between the Catalan numbers Cn, Fine numbers Fn and the Narayana numbers vn;k together with their corresponding generating functions. It is here where we see how the the Lagrange Inversion Formula is applied to complex generating functions to simplify computations. We then study the enumeration of Dyck paths according to the semilength and parameters such as, number of peaks, height of rst peak, number of return steps, e.t.c. We also show how some of these Dyck paths are related. We then make use of Krattenhaler's bijection between 123-avoiding permutations of length n, denoted by Sn(123), and Dyck paths of semilength n. Using this bijective relationship over Sn(123) with k descents and Dyck paths of semilength n with sum of valleys and triple falls equal to k, we get recurrence relationships between ordinary Dyck paths of semilength n and primitive Dyck paths of the same length. From these relationships, we get the generating function for Dyck paths according to semilength, number of valleys and number of triple falls. We nd di erent forms of the generating function for Motzkin paths according to length and number of plateaus with one horizontal step, then extend the discussion to the case where we have more than one horizontal step. We also study Motzkin paths where the horizontal steps have di erent colours, called the k-coloured Motzkin paths and then the k-coloured Motzkin paths which don't have any of their horizontal steps lying on the x-axis, called the k-coloured c-Motzkin paths. We nd that these two types of paths have a special relationship which can be seen from their generating functions. We use this relationship to simplify our enumeration problems

Combinatorics of lattice paths

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014.This dissertation consists of ve chapters which deal with lattice paths such as Dyck paths, skew Dyck paths and generalized Motzkin paths. They never go below the horizontal axis. We derive the generating functions to enumerate lattice paths according to di erent parameters. These parameters include strings of length 2, 3, 4 and r for all r 2 f2; 3; 4; g, area and semi-base, area and semi-length, and semi-base and semi-perimeter. The coe cients in the series expansion of these generating functions give us the number of combinatorial objects we are interested to count. In particular 1. Chapter 1 is an introduction, here we derive some tools that we are going to use in the subsequent Chapters. We rst state the Lagrange inversion formula which is a remarkable tool widely use to extract coe cients in generating functions, then we derive some generating functions for Dyck paths, skew Dyck paths and Motzkin paths. 2. In Chapter 2 we use generating functions to count the number of occurrences of strings in a Dyck path. We rst derive generating functions for strings of length 2, 3, 4 and r for all r 2 f2; 3; 4; g, we then extract the coe cients in the generating functions to get the number of occurrences of strings in the Dyck paths of semi-length n. 3. In Chapter 3, Sections 3.1 and 3.2 we derive generating functions for the relationship between strings of lengths 2 and 3 and the relationship between strings of lengths 3 and 4 respectively. In Section 3.3 we derive generating functions for the low occurrences of the strings of lengths 2, 3 and 4 and lastly Section 3.4 deals with derivations of generating functions for the high occurrences of some strings . 4. Chapter 4, Subsection 4.1.1 deals with the derivation of the generating functions for skew paths according to semi-base and area, we then derive the generating functions according to area. In Subsection 4.1.2, we consider the same as in Section 4.1.1, but here instead of semi-base we use semi-length. The last section 4.2, we use skew paths to enumerate the number of super-diagonal bar graphs according to perimeter. 5. Chapter 5 deals with the derivation of recurrences for the moments of generalized Motzkin paths, in particular we consider those Motzkin paths that never touch the x-axis except at (0,0) and at the end of the path