arXiv subjects

Yahia Djemmada

Publications and source records attributed to Yahia Djemmada.

7 recordsLinked to original sources

arXiv2026

On Deranged Unit-Interval Parking Functions and the Deranged Bell Numbers

Unit-interval parking functions are counted by the Fubini numbers and are in explicit bijection with ordered set partitions. We transport the deranged ordered set partitions of Belbachir, Djemmada, and Németh through this bijection and obtain the deranged unit-interval parking functions $\mathrm{DUPF}_n$. The equality $|\mathrm{DUPF}_n|=\widetilde F_n$, the Stirling-transform formula, the exponential generating function $e^{1-e^x}/(2-e^x)$, and the dominant asymptotics are therefore not presented as new enumerative discoveries; they are consequences of the known deranged Bell-number theory. The new material of this note is the parking-side structure: leader and lucky-car characterizations, a fixed-block stratification of all unit-interval parking functions, rencontres-type generating functions and a Poisson limit law for fixed blocks, a bijective fixed-block decomposition of the Fubini numbers, a multivariate block-size refinement, a fully deranged $r$-start extension, and a Cayley-permutation model based on first appearances.

math.CO↗

arXiv2025

Partial Deranged Bell Numbers and Their Combinatorial Properties

We introduce a novel generalization of deranged Bell numbers by defining the partial deranged Bell numbers $w_{n,r}$, which count the number of set partitions of $\left[ n\right] $ with exactly $r$ fixed blocks, while the remaining blocks are deranged. This construction provides a unified framework that connects partial derangements, Stirling numbers, and ordered Bell numbers. We investigate their combinatorial properties, including explicit formulas, generating functions, and recurrence relations. Moreover, we demonstrate that these numbers are expressible in terms of classical sequences such as deranged Bell numbers and ordered Bell numbers, and reveal their relationship to complementary Bell numbers, offering insights relevant to Wilf's conjecture. Notably, we derive the identity \[ \tildeϕ_{n}=\Tilde{w}_{n,0}-\Tilde{w}_{n,1}=\tilde{w}_{n-1,0}-2\tilde {w}_{n-1,2}, \] which illustrates their structural connection to complementary Bell numbers. We also introduce a polynomial expansion for these numbers and explore their connections with exponential polynomials, geometric polynomials, and Bernoulli numbers. These relationships facilitate the derivation of closed-form expressions for certain finite summations involving Stirling numbers of the second kind, Bernoulli numbers, and binomial coefficients, articulated through partial derangement numbers.

math.CO↗

arXiv2024

The Fibonacci-Fubini and Lucas-Fubini numbers

Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$, we introduce the Fibonacci partition as a Fibonacci permutation of its blocks. Then we define the Fibonacci-Fubini numbers that count the total number of Fibonacci partitions of $[n]$. We study the classical properties of this sequence (generating function, explicit and Dobiński-like formula, etc.), we give combinatorial interpretation, and we extensively examine the Fibonacci-Fubini arithmetic triangle. We give some associate linear recurrence sequences, where in some sequences the Stirling numbers of the first and second kinds appear as well.

math.CO↗

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 deranged Bell numbers

It is known that the ordered Bell numbers count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$. In this paper, we introduce the deranged Bell numbers that count the total number of deranged partitions of $[n]$. We first study the classical properties of these numbers (generating function, explicit formula, convolutions, etc.), we then present an asymptotic behavior of the deranged Bell numbers. Finally, we give some brief results for their $r$-versions.

math.GM↗

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↗

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↗