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Nour-Eddine Fahssi

Publications and source records attributed to Nour-Eddine Fahssi.

6 recordsLinked to original sources
arXiv2026

The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function

This paper studies generalized Pólya conversion problems for the $q$-permanent. We establish a sharp threshold governing the transition from low-dimensional algebraic flexibility to higher-dimensional combinatorial rigidity. For $n \ge 3$ and $q \ne \pm 1$, we prove that the $q$-permanent is not linearly convertible to the determinant or permanent. Conversely, we completely classify the space of Schur multiplier preservers for $n=2$. Focusing on Schur multipliers, we characterize the preserver exponents as a $(2n-2)$-dimensional space of additive matrices. We show that for lower Hessenberg matrices, the general geometric obstruction disappears, yielding an explicit determinantal reduction and an $\mathcal{O}(n^3)$ evaluation algorithm. Furthermore, we classify permutational converter exponents, proving that for $n \ge 4$, admissible symmetries are strictly constrained to the dihedral group. Finally, we resolve a mixed conversion problem, showing the solution space is nonempty only for $n \le 4$, which provides a direct algebraic characterization of the $q$-permanent's zero locus in low dimensions.

math.CO↗
arXiv2019

Fractional Exclusion Statistics as an Occupancy Process

We show the possibility of describing fractional exclusion statistics (FES) as an occupancy process with global and \textit{local} exclusion constraints. More specifically, using combinatorial identities, we show that FES can be viewed as "ball-in-box" models with appropriate weighting on the set of occupancy configurations (merely represented by a partition of the total number of particles). As a consequence, the following exact statement of the generalized Pauli principle is derived: for an $N$-particles system exhibiting FES of extended parameter \mbox{$g=q/r$} ($q$ and $r$ are co-prime integers such that $0 < q \leq r$), (1)~the allowed occupation number of a state is less than or equal to $r-q+1$ and \emph{not} to $1/g$ whenever $q\neq 1$ and (2)~the global occupancy shape is admissible if the number of states occupied by at least two particles is less than or equal to $(N-1)/r$ ($N \equiv 1 \mod r$). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter $g$ and the size $N$.

cond-mat.stat-mech↗
arXiv2018

On the combinatorics of exclusion in Haldane fractional statistics

This paper is a revision of the combinatorics of fractional exclusion statistics (FES). More specifically, the following exact statement of the generalized Pauli principle is derived: for an $N$-particles system exhibiting FES of extended parameter $g=q/r$ ($q$ and $r$ are co-prime integers such that $0 < q \leq r$), we found that the allowed occupation number of a state is smaller than or equal to $r-q+1$ and \emph{not} to $1/g$ whenever $q\neq 1$ and, moreover, the global occupancy shape (merely represented by a partition of $N$) is admissible if the number of states occupied by at least two particles is less than or equal to $(N-1)/r$ ($N \equiv 1 \pmod r$). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter $g$ and the size $N$. As an application of the main result, we study the probability distributions of occupancy configurations. For instance, the number of occupied states is found to be a hypergeometric random variable. Closed-form expressions for the expectation values and variances in the thermodynamic limit are presented. By way of comparison, we obtain parallel results regarding the Gentile intermediate statistics and demonstrate subtle similarities and contrasts with respect to FES.

cond-mat.stat-mech↗
arXiv2016

Some identities involving polynomial coefficients

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several identities and summation formulæ parallel to those of the usual binomial coefficients.

math.NT↗
arXiv2012

Polynomial Triangles Revisited

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to fill this gap. We describe some aspects of these arrays, which generalize similar properties of the binomial coefficients. Some combinatorial models enumerated by polynomial coefficients, including lattice paths model, spin chain model and scores in a drawing game, are introduced. Several known binomial identities are then extended. In addition, we calculate recursively generating functions of column sequences. Interesting corollaries follow from these recurrence relations such as new formulae for the Fibonacci numbers and Hermite polynomials in terms of trinomial coefficients. Finally, properties of the entropy density function that characterizes polynomial coefficients in the thermodynamical limit are studied in details.

math.CO↗