arXiv subjects

El-Mehdi Mehiri

Publications and source records attributed to El-Mehdi Mehiri.

12 recordsLinked to original sources

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

The Parity-Constrained Four-Peg Tower of Hanoi Problem and Its State Graph

We introduce and study a parity-constrained variant of the four-peg Tower of Hanoi problem. In this model, two pegs are neutral, while the two remaining pegs are reserved respectively for even-labelled and odd-labelled discs. Starting from the classical initial tower, we consider four natural transfer objectives corresponding to different target configurations of the full tower and of the even and odd subtowers. For these four objectives, we propose a system of recursive algorithms based on parity separation and classical three-peg transfers. These algorithms lead to a coupled system of recurrence relations for their move counts. The resulting candidate sequences are then transformed into simplified and higher-order recurrences, from which explicit closed formulas are obtained. The formulas exhibit a periodic structure and have the same exponential order of growth, strictly slower than that of the classical three-peg Tower of Hanoi. The main open point is the optimality of the proposed recursive algorithms. Equivalently, one has to prove that certain canonical configurations, including the one-move behaviour of the largest disc, are unavoidable in every shortest solution. This difficulty is closely analogous to the structural difficulties encountered in the Reve's puzzle and the Frame--Stewart conjecture. We therefore formulate the optimality of the proposed algorithms as a conjecture. We also discuss computational evidence, the number of shortest solutions, a linear variant in which only adjacent peg moves are allowed, and the associated state graph of the parity-constrained problem.

math.CO↗

arXiv2026

Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs

We establish a bijective correspondence between Smirnov words with balanced letter multiplicities and Hamiltonian paths in complete $m$-partite graphs $K_{n,n,\ldots,n}$. This bijection allows us to derive closed inclusion-exclusion formulas for the number of Hamiltonian cycles in such graphs. We further extend the enumeration to the generalized nonuniform case $K_{n_1,n_2,\ldots,n_m}$. We also provide an asymptotic analysis based on Stirling's approximation, which yields compact factorial expressions and logarithmic expansions describing the growth of the number of Hamiltonian cycles in the considered graphs.

math.CO↗

arXiv2026

The Weighted Tower of Hanoi: Algebraic Structure, Phase Transitions, and Integer Sequences

We develop a unified algebraic theory of the weighted Tower of Hanoi with arbitrary nonnegative symmetric move costs depending on both disc index and pegs. Starting from a general optimality recurrence with two competing strategies -- one largest-disc move (one-LDM) and two largest-disc moves (two-LDM) -- we derive complete matrix formulations for both regimes and obtain explicit closed forms for the minimal transfer cost. The one-LDM dynamics is governed by a nontrivial linear operator whose spectral decomposition reveals a fundamental connection with the Jacobsthal and Lichtenberg sequences, while the two-LDM dynamics exhibits pure exponential growth. This framework yields exact solutions for broad classes of weight models, including peg-symmetric, disc-symmetric, polynomial, geometric, arithmetic, and sequence-induced costs. In particular, choosing classical integer sequences (Fibonacci, Lucas, Jacobsthal, Pell, Euler, etc.) as disc weights produces new derived sequences with explicit formulas and recurrences, establishing the Tower of Hanoi as a sequence-generating transform. We further introduce and analyze models with forbidden moves and move-type-dependent weights, uncovering a phase transition phenomenon in which the optimal strategy switches from two-LDM behavior for small discs to one-LDM behavior beyond a finite threshold. Our results provide a comprehensive algebraic and combinatorial understanding of weighted Hanoi dynamics and expose deep connections between optimal solutions and classical integer sequences.

math.CO↗

arXiv2026

The Power Contamination Problem on Grids Revisited: Optimality, Combinatorics, and Links to Integer Sequences

This paper presents a combinatorial study of the power contamination problem, a dynamic variant of power domination modeled on grid graphs. We resolve a conjecture posed by Ainouche and Bouroubi (2021) by proving it is false and instead establish the exact value of the power contamination number on grid graphs. Furthermore, we derive recurrence relations for this number and initiate the enumeration of optimal contamination sets. We prove that the number of optimal solutions for specific grid families corresponds to well-known integer sequences, including those counting ternary words with forbidden subwords and the large Schröder numbers. This work settles the fundamental combinatorial questions of the power contamination problem on grids and reveals its rich connections to classical combinatorics.

math.CO↗

arXiv2026

M-Polynomial of Product Graphs

The M-polynomial provides a unifying framework for a wide class of degree-based topological indices. Despite its structural importance, general methods for computing the M-polynomial under graph constructions remain limited. In this paper, explicit formulas, and compact ones whenever possible, for the M-polynomial under different graph products whose vertex sets are the Cartesian product of the factors are developed. The products studied are the direct, the Cartesian, the strong, the lexicographic, the symmetric-difference, the disjunction, and the Sierpiński product. The obtained formulas yield a unified structural description of how vertex-degree interactions propagate under graph constructions and extend existing results for degree-based indices at the polynomial level.

math.CO↗

arXiv2026

Block-Separated Overpartitions: Fibonacci Structure and Euler Factorization

We introduce and study block-separated overpartitions, a constrained family of overpartitions in which no two consecutive distinct part-blocks are both overlined. This local restriction produces a new sequence that naturally interpolates between classical partitions and unrestricted overpartitions. We show that the internal decoration of distinct part-blocks is governed by Fibonacci-type combinatorics: once the set of distinct part-sizes is fixed, the admissible overlining patterns are counted by Fibonacci numbers. This leads to a symmetric-function expansion of the generating function and a two-state transfer-matrix formulation. After extracting the Euler product, we obtain normalized recurrences, second-order scalar recurrences, determinantal representations, and a continued-fraction description of finite truncations. Finally, we determine the asymptotic growth of the counting function, and prove that block-separated overpartitions share the same exponential scale as ordinary partitions, with a modified subexponential constant.

math.CO↗

arXiv2025

Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs

The M-polynomial, introduced by Deutsch and Klavžar in 2015, provides a unifying algebraic framework for the computation of numerous degree-based topological indices such as the Zagreb, Randic, harmonic, and forgotten indices. Despite its broad applications in chemical graph theory and network analysis, closed expressions of the M-polynomial remain unknown for many important graph families. In this work we derive, for the first time, a complete explicit expression of the M-polynomial of the generalized Hanoi graphs $H_p^n$ for arbitrary positive $p$ and $n$. Our derivation relies on a detailed combinatorial analysis of the occupancy-based structure of $H_p^n$, refined using Stirling and $2$-associated Stirling numbers to enumerate all configurations with prescribed singleton and multiton counts. We obtain closed formulas for all diagonal and off-diagonal coefficients of the M-polynomial and show how these expressions yield exact values of the main degree-based topological indices. The correctness of the formulas is supported through numerical computation in small instances. These results provide a complete degree-based description of $H_p^n$ and make their structural complexity fully accessible through the M-polynomial framework.

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↗

arXiv2023

On the restricted Hanoi Graphs

Consider the restricted Hanoi graphs which correspond to the variants of the famous Tower of Hanoi problem with multiple pegs where moves of the discs are restricted throughout the arcs of a movement digraph whose vertices represent the pegs of the puzzle and an arc from vertex $p$ to vertex $q$ exists if and only if moves from peg $p$ to peg $q$ are allowed. In this paper, we gave some notes on how to construct the restricted Hanoi graphs as well as some combinatorial results on the number of arcs in these 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↗