arXiv subjects

Hacène Belbachir

Publications and source records attributed to Hacène Belbachir.

At least 19 recordsLinked to original sources

arXiv2026

Circular s-choice parking functions: an exact closed formula via rotational symmetry

We study a circular variant of the $s$-choice parking model: $n$ cars park on $m=n+1$ spots arranged on a circle, each car carrying an anchor and $s-1$ clockwise increments at least $d$ apart with return gap at least $d$; a car tries its choices in order and then sweeps clockwise from its last choice. On the circle every car parks and exactly one spot remains empty. Exploiting rotational symmetry in the spirit of Pollak's proof of the count $(n+1)^{n-1}$, we prove that the empty spot is exactly equidistributed, which yields the closed formula $m^{n-1}\binom{m-sd+s-1}{s-1}^{n}$ for the number of preferences leaving any prescribed spot empty. This appears to be the first closed product formula in the multi-choice parking landscape. We further prove a refinement: within every class of preferences with prescribed increments, the empty spot is still exactly equidistributed, which explains the product structure of the formula and, for $s=2$, $d=1$, the appearance of the classical count $(n+1)^{n-1}$ as a factor of $(n+1)^{n-1}n^{n}$. The admissible tuples are enumerated through Kaplansky's lemma on circular selections, and all results are verified by exhaustive computer enumeration.

math.CO↗

arXiv2026

Rotating-Memory Fibonacci Numbers and Periodic Tilings

We introduce and study the rotating-memory Fibonacci numbers, a periodic variable-order analogue of the Fibonacci sequence in which the number of preceding terms used in the recurrence changes cyclically with the index. Despite this varying memory, the resulting sequences exhibit a remarkably rigid structure. We derive closed forms, rational generating functions, arithmetic properties, and exact growth behavior, and show that the sequence decomposes naturally into geometric subsequences. We also develop combinatorial interpretations in terms of periodically constrained tilings, and restricted compositions, including bijective explanations for the multiplicative structure of the sequence. In addition, the first two nonclassical periods admit natural geometry-driven realizations: the period-2 sequence arises from monomer--dimer tilings of a triangular chain, while the period-3 sequence is related to tilings of a double hexagon strip by single and double hexagons. These connections provide geometric interpretations of the rotating recurrence in which the periodic behavior is induced by the underlying structures themselves, and suggest a broader problem of constructing analogous models for higher periods.

math.CO↗

arXiv2026

Generalized Steinhaus triangles generated by canonical basis vectors: periodicity and weight formulas

A generalized Steinhaus $s$-triangle is obtained from a binary sequence by repeatedly replacing each block of $s$ consecutive entries by its sum modulo $2$. We study the triangles generated by canonical basis vectors. Expressing their entries through bi$^{s}$nomial coefficients, we prove that the truncated coefficient profiles of the successive rows are purely periodic and determine their exact period. This periodicity yields a power-of-two derivative identity and a decomposition into identical finite blocks. We consequently obtain an exact recurrence for the number of ones and show that, on every residue class, this weight is an affine function of the length. We also derive a rational generating function, determine the asymptotic growth rate, and prove that these canonical triangles have zero density of ones. Explicit formulas are obtained for the first two canonical vectors and for the block weight in the classical case. These results provide a unified description of the periodic and enumerative structure of generalized Steinhaus triangles generated by canonical basis vectors.

math.CO↗

arXiv2026

Delannoy--Steinhaus triangles over $\mathbb{Z}/2\mathbb{Z}$: weight spectrum, balanced triangles, and extremal values

A Delannoy--Steinhaus triangle is obtained from a finite sequence by a recurrence governed by the Delannoy numbers. We introduce this construction over $\mathbb{Z}/2\mathbb{Z}$ and study its weight distribution. The relevant Delannoy coefficients are all odd, which reduces every entry to the parity of a consecutive interval of the generating sequence. Encoding these interval parities by prefix parities yields a weight formula depending only on the numbers of zeros and ones in the prefix-parity sequence. We use this formula to determine the complete weight spectrum and the exact multiplicity of each weight. As a consequence, we characterize and enumerate the balanced triangles: a balanced triangle generated by a binary sequence of length $n$ exists if and only if $n+1$ is a perfect square. We also determine the canonical-vector weights, the minimum nonzero weight, the {second-smallest nonzero weight}, the maximum weight, and the average weight.

math.CO↗

arXiv2026

The distribution of the de Moivre experiment

In this paper, we focus on the de Moivre random experiment, which allows us to introduce the $s$-Bernoulli distribution and the bi$^{s}$nomial distribution. We present some probabilistic properties such as the expectation, the variance, the skewness and kurtosis coefficients, the moments, the cumulants and the generating functions; the moments are expressed through extended Eulerian numbers and partial Bell polynomials, and the cumulants through logarithmic polynomials. Then we establish that, for an integer $s\geq 2$, the bi$^{s}$nomial distribution converges to a Poisson limiting distribution when the expectations converge, and that, for fixed parameters with $p\neq q$, it satisfies a central limit theorem as $n\rightarrow\infty$.

math.PR↗

arXiv2026

A random approach to the multibonacci sequence

This paper presents a random approach to the multibonacci sequence. We generalise the model introduced by Benjamin, Levin, Mahlburg, and Quinn, which is based on a random tiling method using dominoes and squares that leads to the Fibonacci sequence, and which was extended to the tribonacci case in a previous work by the authors. Our approach employs tiling with linear $k$-ominoes, $k=1,\ldots,s$, combined with specific colouring, to generate a weighted multibonacci sequence. For a natural random variable~$X$ defined by this model, we establish the distribution of $X$ in terms of multibonacci numbers and compute $\mathbb{E}[X] = 2^{s+1}-3$.

math.CO↗

arXiv2025

The Towers of Fibonacci, Lucas, Pell, and Jacobsthal

We present in this paper four new variants of the Tower of Hanoi problem, the optimal solution of each of these variants is related to one of the four known numbers Fibonacci, Lucas, Pell, and Jacobsthal. We give an optimal solution to each of these variants, and we present their associated graphs.

math.CO↗

arXiv2022

Enumerating moves in the optimal solution of the Tower of Hanoi

In the Tower of Hanoi problem, there is six types of moves between the three pegs. The main purpose of the present paper is to find out the number of each of these six elementary moves in the optimal sequence of moves. We present a recursive function based on indicator functions, which counts the number of each elementary move, we investigate some of its properties including combinatorial identities, recursive formulas and generating functions. Also we found and interesting sequence that is strongly related to counting each type of these elementary moves that we'll establish some if its properties as well.

math.CO↗

arXiv2022

The weighted Tower of Hanoi

The weighted Tower of Hanoi is a new generalization of the classical Tower of Hanoi problem, where a move of a disc between two pegs $i$ and $j$ is weighted by a positive real $w_{ij}\geq 0$. This new problem generalizes the concept of finding the minimum number of moves to solve the Tower of Hanoi, to find a sequence of moves with the minimum total cost. We present an optimal dynamic algorithm to solve the weighted Tower of Hanoi problem, we also establish some properties of this problem, as well as its relation with the Tower of Hanoi variants that are based on move restriction.

cs.DM↗

arXiv2021

Enumeration of parallelogram polycubes

In this paper, we enumerate parallelogram polycubes according to several parameters. After establishing a relation between Multiple Zeta Function and the Dirichlet generating function of parallelogram polyominoes, we generalize it to the case of parallelogram polycubes. We also give an explicit formula and an ordinary generating function of parallelogram polycubes according to the width, length and depth, by characterizing its projections. Then, these results are generalized to polyhypercubes.

cs.DM↗

arXiv2021

Unified Bernoulli-Euler polynomials of Apostol type

The object of this paper is to introduce and study properties of unified Apostol-Bernoulli and Apostol-Euler polynomials noted by $\left\{\mathfrak{V_{n}}(x;λ;μ)\right\}_{n \geq 0}$. We study some arithmetic properties of $\left\{\mathfrak{V_{n}}(x;λ;μ)\right\}_{n \geq 0}$ as their connection to Apostol-Euler polynomials and Apostol-Bernoulli polynomials. Also, we give derivation and integration representations of $\left\{\mathfrak{V_{n}}(x;λ;μ)\right\}_{n \geq 0}$. Finally, we use the umbral calculus approach to deduce symmetric identities.

math.CO↗

arXiv2021

The $(l,r)$-Stirling numbers: a combinatorial approach

This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations and symmetric functions. Also, we give a limit representation of the multiple zeta function using $(l,r)$-Stirling of the first kind.

math.CO↗

arXiv2020

On the attractor zero of a sequence of polynomials

The main perpose of this paper is to sudy the roots of a familly of polynomials that arise from a linear recurrences associated to Pascal's triangle and their zero attractor, using an analytical methods based on conformal mappings.

math.CA↗

arXiv2020

Asymmetric extension of Pascal-Dellanoy triangles

We give a generalization of the Pascal triangle called the quasi s-Pascal triangle where the sum of the elements crossing the diagonal rays produce the s-bonacci sequence. For this, consider a lattice path in the plane whose step set is {L = (1, 0), L1 = (1, 1), L2 = (2, 1), . . . , Ls = (s, 1)}; an explicit formula is given. Thereby linking the elements of the quasi s-Pascal triangle with the bisnomial coefficients. We establish the recurrence relation for the sum of elements lying over any finite ray of the quasi s-Pascal triangle. The generating function of the cited sums is produced. We also give identities among which one equivalent to the de Moivre sum and establish a q-analogue of the coefficient of the quasi s-Pascal triangle.

math.CO↗

arXiv2019

Directed Plateau Polyhypercubes

In this paper, we study a particular family of polyhypercubes in dimension $d\geq 3$, the directed plateau polyhypercubes, according to to the width and a new parameter the lateral area. We give an explicit formula and we also propose an expression of the generating function in this case.

math.CO↗

arXiv2018

On central Fubini-like numbers and polynomials

We introduce the central Fubini-like numbers and polynomials using Rota approach. Several identities and properties are established as generating functions, recurrences, explicit formulas, parity, asymptotics and determinantal representation.

math.CO↗

arXiv2018

Plateau Polycubes and Lateral Area

In this paper, we enumerate two families of polycubes, the directed plateau polycubes and the plateau polycubes, with respect to the width and a new parameter, the Lateral Area. We give an explicit formula and the generating function for each of the two families of polycubes. Moreover, some asymptotic results about plateau polycubes are provided. We also establish results concerning the enumeration of column-convex polyominoes that are useful to get asymptotic results of polycubes.

math.CO↗

arXiv2017

On the self-convolution of generalized Fibonacci numbers

We focus on a family of equalities pioneered by Zhang and generalized by Zao and Wang and hence by Mansour which involves self convolution of generalized Fibonacci numbers. We show that all these formulas are nicely stated in only one equation involving a bivariate ordinary generating function and we give also a formula for the coefficients appearing in that context. As a consequence, we give the general forms for the equalities of Zhang, Zao-Wang and Mansour.

math.CO↗