arXiv subjects

Toufik Mansour

Publications and source records attributed to Toufik Mansour.

At least 55 records · Page 3Linked to original sources

arXiv2019

On typical triangulations of a convex $n$-gon

Let $f_n$ be a function assigning weight to each possible triangle whose vertices are chosen from vertices of a convex polygon $P_n$ of $n$ sides. Suppose ${\mathcal T}_n$ is a random triangulation, sampled uniformly out of all possible triangulations of $P_n$. We study the sum of weights of triangles in ${\mathcal T}_n$ and give a general formula for average and variance of this random variable. In addition, we look at several interesting special cases of $f_n$ in which we obtain explicit forms of generating functions for the sum of the weights. For example, among other things, we give new proofs for already known results such as the degree of a fixed vertex and the number of ears in ${\mathcal T}_n,$ as well as, provide new results on the number of "blue" angles and refined information on the distribution of angles at a fixed vertex. We note that our approach is systematic and can be applied to many other new examples while generalizing the existing results.

math.CO↗

arXiv2019

Horizontal visibility graph of a random restricted growth sequence

We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size $n.$ Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers.

math.CO↗

arXiv2019

On the Complementary Equienergetic Graphs

Energy of a simple graph $G$, denoted by $\mathcal{E}(G)$, is the sum of the absolute values of the eigenvalues of $G$. Two graphs with the same order and energy are called equienergetic graphs. A graph $G$ with the property $G\cong \overline{G}$ is called self-complementary graph, where $\overline{G}$ denotes the complement of $G$. Two non-self-complementary equienergetic graphs $G_1$ and $G_2$ satisfying the property $G_1\cong \overline{G_2}$ are called complementary equienergetic graphs. Recently, Ramane et al. [Graphs equienergetic with their complements, MATCH Commun. Math. Comput. Chem. 82 (2019) 471-480] initiated the study of the complementary equienergetic regular graphs and they asked to study the complementary equienergetic non-regular graphs. In this paper, by developing some computer codes and by making use of some software like Nauty, Maple and GraphTea, all the complementary equienergetic graphs with at most 10 vertices as well as all the members of the graph class $Ω=\{G \ : \ \mathcal{E}(L(G)) = \mathcal{E}(\overline{L(G)}) \text{, the order of $G$ is at most 10}\}$ are determined, where $L(G)$ denotes the line graph of $G$. In the cases where we could not find the closed forms of the eigenvalues and energies of the obtained graphs, we verify the graph energies using a high precision computing (2000 decimal places) of Maple. A result about a pair of complementary equienergetic graphs is also given at the end of this paper.

math.CO↗

arXiv2019

Height of records in partitions of a set

We study the restricted growth function associated with set partitions, and obtain exact formulas for the number of strong records with height one, the total of record heights over set of partitions, and the number of partitions with a given maximal height of strong records. We also extend some of these results to weak records.

math.CO↗

arXiv2019

Staircase patterns in words: subsequences, subwords, and separation number

We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number, the latter being defined as the number of consecutive maximal staircase subwords packed in a word. We study asymptotic properties of the sequence $h_{r,k}(n),$ the number of $n$-array words with $r$ separations over alphabet $[k]$ and show that for any $r\geq 0,$ the growth sequence $\big(h_{r,k}(n)\big)^{1/n}$ converges to a characterized limit, independent of $r.$ In addition, we study the asymptotic behavior of the random variable $\mathcal{S}_k(n),$ the number of staircase separations in a random word in $[k]^n$ and obtain several limit theorems for the distribution of $\mathcal{S}_k(n),$ including a law of large numbers, a central limit theorem, and the exact growth rate of the entropy of $\mathcal{S}_k(n).$ Finally, we obtain similar results, including growth limits, for longest $L$-staircase subwords and subsequences.

math.CO↗

arXiv2019

On ballistic deposition process on a strip

We revisit the model of the ballistic deposition studied in \cite{bdeposition} and prove several combinatorial properties of the random tree structure formed by the underlying stochastic process. Our results include limit theorems for the number of roots and the empirical average of the distance between two successive roots of the underlying tree-like structure as well as certain intricate moments calculations.

math.PR↗

arXiv2019

Finite automata, probabilistic method, and occurrence enumeration of a pattern in words and permutations

The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given pattern $v$ exactly $r$ times. In addition, we study the asymptotic behavior of the random variable $X_n,$ the number of pattern occurrences in a random $n$-array word. The two topics are closely related through the identity $P(X_n=r) = $ $\frac{1}{k^n}f_r^v(k,n).$ In particular, we show that for any $r\geq 0,$ the Stanley-Wilf sequence $\bigl(f_r^v(k,n)\bigr)^{1/n}$ converges to a limit independent of $r,$ and determine the value of the limit. We then obtain several limit theorems for the distribution of $X_n,$ including a CLT, large deviation estimates, and the exact growth rate of the entropy of $X_n.$ Furthermore, we introduce a concept of weak avoidance and link it to a certain family of non-product measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley-Wilf sequences, to pattern occurrences in permutations.

math.CO↗

arXiv2018

$q$-deformed conformable fractional Natural transform

In this paper, we develop a new deformation and generalization of the Natural integral transform based on the conformable fractional $q$-derivative. We obtain transformation of some deformed functions and apply the transform for solving linear differential equation with given initial conditions.

math.CA↗

arXiv2018

New degenerated polynomials arising from non-classical Umbral Calculus

We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present generalizations of some familiar identities and connection between these kinds of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish a new analogue of the Euler identity for the degenerate Bernoulli numbers.

math.NT↗

arXiv2018

Passing through a stack $k$ times with reversals

We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the $2$-reverse pass sortable permutations and give computational results for some classes with larger maximal rev-tier. We also show all classes of $(t+1)$-reverse pass sortable permutations are finitely based. Additionally, a new Entringer family consisting of maximal rev-tier permutations of length $n$ was discovered along with a bijection between this family and the collection of alternating permutations of length $n-1$. We calculate generating functions for the number permutations of length $n$ and exact rev-tier $t$.

math.CO↗

arXiv2018

Passing through a stack $k$ times

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $π$ to be $k$-pass sortable if $π$ is sortable using $k$ passes through the stack. Permutations that are $1$-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of $2$-pass sortable permutations in terms of their basis. We also show all $k$-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer $k$. Finally, we define the notion of tier of a permutation $π$ to be the minimum number of passes after the first pass required to sort $π$. We then give a bijection between the class of permutations of tier $t$ and a collection of integer sequences studied by Parker. This gives an exact enumeration of tier $t$ permutations of a given length and thus an exact enumeration for the class of $(t+1)$-pass sortable permutations. Finally, we give a new derivation for the generating function in Parker's thesis and an explicit formula for the coefficients.

math.CO↗

arXiv2018

The Twelvefold way, the non-intersecting circles problem, and partitions of multisets

Let $n$ be a non-negative integer and $A=\{a_1,\ldots,a_k\}$ be a multi-set with $k$ not necessarily distinct members, where $a_1\leqslant\ldots\leqslant a_k$. We denote by $Δ(n,A)$ the number of ways to partition $n$ as the form $a_1x_1+\ldots+a_kx_k$, where $x_i$'s are distinct positive integers and $x_i< x_{i+1}$ whenever $a_i=a_{i+1}$. We give a recursive formula for $Δ(n,A)$ and some explicit formulas for some special cases. Using this notion we solve the non-intersecting circles problem which asks to evaluate the number of ways to draw $n$ non-intersecting circles in a plane regardless to their sizes. The latter also enumerates the number of unlabelled rooted tree with $n+1$ vertices.

math.CO↗

arXiv2018

Several variants of the Dumont differential system and permutation statistics

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (Math Comp, 1979, 33: 1293--1297) and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi-pairs of differential equations. We present a general method to derive the solutions of these differential equations. As applications, we present some characterizations for several permutation statistics.

math.CO↗

arXiv2017

Enumeration of permutations avoiding a triple of 4-letter patterns is all done

This paper completes a project to enumerate permutations avoiding a triple T of 4-letter patterns, in the sense of classical pattern avoidance, for every T. There are 317 symmetry classes of such triples T and previous papers have enumerated avoiders for all but 14 of them. One of these 14 is conjectured not to have an algebraic generating function. Here, we find the generating function for each of the remaining 13, and it is algebraic in each case.

math.CO↗

arXiv2017

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.

math.CO↗

arXiv2017

Enumeration of small Wilf classes avoiding 1342 and two other $4$-letter patterns

This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.

math.CO↗

arXiv2016

Restricted Stirling permutations

In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and explicit formulas are studied. Combinatorial explanations are given for some equidistributions.

math.CO↗

arXiv2016

On moments of the integrated exponential Brownian motion

We present new exact expressions for a class of moments for the geometric Brownian motion, in terms of determinants, obtained using a recurrence relation and combinatorial arguments for the case of a Ito's Wiener process. We then apply the obtained exact formulas to computing averages of the solution of the logistic stochastic differential equation via a series expansion, and compare the results to the solution obtained via Monte Carlo.

cond-mat.stat-mech↗