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Toufik Mansour

Publications and source records attributed to Toufik Mansour.

At least 19 recordsLinked to original sources
arXiv2026

Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. I: Algebraic Framework and Combinatorial Identities

The $(p,q)$-deformed generalized Weyl algebra is generated by variables $X, Y$ and $Z_p$ which satisfy the commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. We investigate the problem of normal ordering arbitrary words in these letters with the help of Young diagrams, and we treat certain special cases explicitly. In particular, the connection to generalized Stirling numbers is considered in detail.

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arXiv2026

Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. II: Interpretation in terms of rook placements

In this paper, we investigate the combinatorial structure arising from the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$, and $Z_p$, satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, where $s\in \mathbb{N}_0$. Our primary objective is to use the normal ordering process defined by these relations to develop a novel model of $(p, q)$-deformed rook theory. Specifically, we introduce a new framework of $(p, q)$-deformed $s$-rook numbers derived from this normal ordering process. Utilizing these combinatorial models, we provide explicit combinatorial interpretations for the associated $(p, q)$-generalized Stirling numbers via rook placements on staircase boards. Our results extend several classical and recent formulations in the literature to the general $p\neq 1$ setting.

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arXiv2026

Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. III: The binomial formula

We study the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$ and $Z_p$ satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. Within this framework, we investigate the noncommutative binomial formula $(X+Y)^n$ and related identities. In particular, we show how the associated normal ordering coefficients can be expressed in terms of $(p,q)$-deformed $s$-rook numbers. We treat several special cases explicitly, recovering known results from literature as well as deriving new ones.

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arXiv2025

On ascent sequences avoiding 021 and a pattern of length four

Ascent sequences of length $n$ avoiding the pattern $021$ are enumerated by the $n$-th Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$. In this paper, we extend this result and enumerate ascent sequences avoiding $\{021,τ\}$, where $τ$ is a pattern of length four. We in turn identify all of the corresponding Wilf-equivalence classes and find generating function formulas corresponding to each class. In a couple of cases, we make use of an auxiliary statistic and the kernel method to ascertain the generating function. In several cases, our work of enumeration is shortened by establishing the equivalence of $\{021,τ\}$- and $\{021,τ'\}$-avoiders of a given length through an explicit bijection. As a consequence of our results, one obtains new combinatorial interpretations in terms of ascent sequences for several of the entries in the OEIS.

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arXiv2025

Avoidance of vincular patterns by flattened derangements

In this paper, we consider the problem of avoiding a single vincular pattern of length three by derangements in the flattened sense and find explicit formulas for the generating functions enumerating members of each corresponding avoidance class according to the number of cycles. We make frequent use of the kernel method in solving the functional equations that arise which are satisfied by these (ordinary) generating functions. In the case of avoiding 23-1, which is equivalent to 32-1 in the flattened sense, it is more convenient to consider the exponential generating function instead due to the form of the recurrence. This leads to an explicit expression for the distribution of the number of cycles in terms of Stirling numbers of the second kind and the determinant of a certain tridiagonal matrix. Finally, the cases of 3-12 and 3-21 are perhaps the most difficult of all, and here we make use of a pair of auxiliary statistics in order to find a system of recurrences that enumerate each avoidance class.

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arXiv2025

Counting k-ary words by number of adjacency differences of a prescribed size

Recently, the general problem of enumerating permutations $π=π_1\cdots π_n$ such that $π_{i+r}-π_i \neq s$ for all $1\leq i\leq n-r$, where $r$ and $s$ are fixed, was considered by Spahn and Zeilberger. In this paper, we consider an analogous problem on $k$-ary words involving the distribution of the corresponding statistic. Note that for $k$-ary words, it suffices to consider only the $r=1$ case of the aforementioned problem on permutations. Here, we compute for arbitrary $s$ an explicit formula for the ordinary generating function for $n \geq 0$ of the distribution of the statistic on $k$-ary words $ρ=ρ_1\cdotsρ_n$ recording the number of indices $i$ such that $ρ_{i+1}-ρ_i=s$. This result may then be used to find a comparable formula for finite set partitions with a fixed number of blocks, represented sequentially as restricted growth functions. Further, several sequences from the OEIS arise as enumerators of certain classes of $k$-ary words avoiding adjacencies with a prescribed difference. The comparable problem where one tracks indices $i$ such that the absolute difference $|a_{i+1}-a_i|$ is a fixed number is also considered on $k$-ary words and the corresponding generating function may be expressed in terms of Chebyshev polynomials. Finally, combinatorial proofs are found for several related recurrences and formulas for the total number of adjacencies of the form $a(a+s)$ on the various structures.

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arXiv2025

Sorting inversion sequences

We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and generalizations, some of which we apply here. We give the classification of several classes of sortable inversion sequences in terms of pattern avoidance. We are able to provide an exact enumeration of some of the sortable classes in question using both classical approaches and a more recent strategy utilizing generating trees.

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arXiv2024

Pattern Avoidance in Weak Ascent Sequences

In this paper, we study pattern avoidance in weak ascent sequences, giving some results for patterns of length 3. This is an analogous study to one given by Duncan and Steingrímsson (2011) for ascent sequences. More precisely, we provide systematically the generating functions for the number of weak ascent sequences avoiding the patterns $001, 011, 012, 021$, and $102$. Additionally, we establish bijective connections between pattern-avoiding weak ascent sequences and other combinatorial objects, such as compositions, upper triangular 01-matrices, and plane trees.

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arXiv2024

A curious symmetric decomposition of the (des, exc)-Eulerian polynomials

One of the most central result in combinatorics says that the descent statistic and the excedance statistic are equidistribued over the symmetric group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6) (2010), 2921--2966), we provide a curious $t$-symmetric decomposition for the generating polynomial of the joint distribution of the descent and excedance statistics over the symmetric group.

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arXiv2024

Avoidance of vincular patterns by Catalan words

Let $\mathcal{C}_n$ denote the set of words $w=w_1\cdots w_n$ on the alphabet of positive integers satisfying $w_{i+1}\leq w_i+1$ for $1 \leq i \leq n-1$ with $w_1=1$. The members of $\mathcal{C}_n$ are known as Catalan words and are enumerated by the $n$-th Catalan number $C_n$. The problem of finding the cardinality of various avoidance classes of $\mathcal{C}_n$ has been an ongoing object of study, and members of $\mathcal{C}_n$ avoiding one or two classical or a single consecutive pattern have been enumerated. In this paper, we extend these results to vincular patterns and seek to determine the cardinality of each avoidance class corresponding to a pattern of type (1,2) or (2,1). In several instances, a simple explicit formula for this cardinality may be given. In the more difficult cases, we find only a formula for the (ordinary) generating function which enumerates the class in question. We make extensive use of functional equations in establishing our generating function results.

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arXiv2024

Counting $r\times s$ rectangles in (Catalan) words

Generalizing previous results, we introduce and study a new statistic on words, that we call rectangle capacity. For two fixed positive integers $r$ and $s$, this statistic counts the number of occurrences of a rectangle of size $r\times s$ in the bargraph representation of a word. We find the bivariate generating function for the distribution on words of the number of $r\times s$ rectangles and the generating function for their total number over all words. We also obtain the analog results for Catalan words.

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arXiv2024

Normal ordered grammars

We introduce the theory of normal ordered grammars, which gives a natural generalization of the normal ordering problem. To illustrate the main idea, we explore normal ordered grammars associated with the Eulerian polynomials and the second-order Eulerian polynomials. In particular, we present a normal ordered grammatical interpretation for the (cdes,cyc) (p,q)-Eulerian polynomials, where cdes and cyc are the cycle descent and cycle statistics, respectively. The exponential generating function for a family of polynomials, generated by a normal ordered grammar associated with the second-order Eulerian polynomials, reveals an interesting feature: its expression involves the generating function for Catalan numbers as its exponent. In the final part, we discuss some normal ordered grammars related to the type B Eulerian polynomials. A normal ordered grammatical interpretation of the up-down run polynomial is also established.

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arXiv2023

Staircase graph words

Generalizing the notion of staircase words, introduced by Knopfmacher et.\ al, we define staircase graph words. These are functions $w$ from the vertex set $V$ of a graph into the set $\{1,2,\ldots,k\}$, such that $|w(x)-w(y)|\leq 1$, for every adjacent $x,y\in V$. We find the explicit generating functions for the number of staircase graph words for the grid graph, the rectangle-triangular graph and the king's graph, all of size $2\times n$.

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arXiv2023

An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences

We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern-classes $I_n(000, 021), I_n(100, 021)$, $I_n(110, 021), I_n(102, 021)$, $I_n(100,012)$, $I_n(011,201)$, $I_n(011,210)$ and $I_n(120,210)$. Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes.

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arXiv2023

Random Walk Labelings of Perfect Trees and Other Graphs

A Random walk labeling of a graph $G$ is any labeling of $G$ that could have been obtained by performing a random walk on $G$. Continuing two recent works, we calculate the number of random walk labelings of perfect trees, combs, and double combs, the torus $C_2\times C_n$, and the graph obtained by connecting three path graphs to form two cycles.

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arXiv2023

On the maximal sum of the entries of a matrix power

Let $p_n$ be the maximal sum of the entries of $A^2$, where $A$ is a square matrix of size $n$, consisting of the numbers $1,2,\ldots,n^2$, each appearing exactly once. We prove that $m_n=Θ(n^7)$. More precisely, we show that $n(240n^{6}+28n^{5}+364n^{4}+210n^{2}-28n+26-105((-1)^{n}+1))/840\leq p_n\leq n^{3}(n^{2}+1)(7n^{2}+5)/24$.

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arXiv2023

Further Results on Random Walk Labelings

Recently, we initiated the study of random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that each vertex is labeled upon its first visit. In this work, we calculate the number of random walk labelings of several natural graph families: The wheel, fan, barbell, lollipop, tadpole, friendship, and snake graphs. Additionally, we prove several combinatorial identities that emerged during the calculations.

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