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Madjid Mirzavaziri

Publications and source records attributed to Madjid Mirzavaziri.

18 recordsLinked to original sources
arXiv2026

Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler

We investigate Conjecture 7.4 of Guo and Krattenthaler \cite{GK}, which predicts that for every positive integer $n$, the polynomial \[ P_n(q) = \frac{(1-q)^2}{(1-q^{10n-1})(1-q^{15n-1})}\qbin{30n}{5n} \] has non-negative coefficients, except for $q^1$ and $q^{125n^2-25n+3}$, which equal $-1$. We study this problem through three main perspectives: First, by linking the boundary coefficients to the second differences of the partition function, we settle the first and last $25n+1$ coefficients via a novel partition injection. Second, by introducing a ``defect'' generalization of Andrews' classical lemma, we reduce the conjecture to an almost-unimodality property and employ a uniform saddle-point analysis to prove it analytically for all large $n \ge 2\cdot 10^{29}$. Finally, we connect these quotients to graph binomial coefficients, providing a natural orbit-counting interpretation via Burnside's lemma on directed cycles and paths.

math.CO↗
arXiv2026

Riordan Arrays and Shifted Hankel Determinants of OEIS A005773

We study two Riordan arrays associated with the Motzkin-prefix sequence (\href{https://oeis.org/A005773}{OEIS A005773}). By connecting their production matrices through a simple boundary factorization, we obtain a period-three transformation on Motzkin endpoint distributions and exact formulas for paths by final height modulo~$3$. In addition, a unified factorization for the shifted Hankel matrices leads directly to explicit closed forms for the third and fourth Hankel determinants, $Δ_n^{(3)}$ and~$Δ_n^{(4)}$.

math.CO↗
arXiv2026

Cyclic Latin Eulerian Numbers

We introduce the directed cyclic difference inventory $D_n(\mathbf{m})$ to study Latin Eulerian numbers restricted to row-reorderings of the cyclic Latin square, offering an orientation-sensitive refinement of the prescribed-edge-length Hamiltonian path problem. We establish exact enumeration formulas, symmetries, and realizability obstructions for this inventory. Furthermore, we reduce the cyclic total-ascent statistic directly to endpoint-refined Eulerian statistics via the identity $Σ(L_π) = n \operatorname{asc}(π) + π(1) - π(n)$, yielding a closed-form expression for the cyclic Latin-Eulerian polynomial.

math.CO↗
arXiv2026

A Combinatorial Problem in Cinema Seating

We address a problem concerning cinema audiences: ``A cinema has $n$ seats numbered from $1$ to $n$, and there are $n$ people with tickets numbered from $1$ to $n$. People enter the cinema in order. If someone has the ticket number $i$, they can choose seats whose numbers are multiples of $i$. They should exit the cinema if the permitted seats are occupied by previous audience members. In how many ways can they be seated under these conditions?" We give an algorithm to create the list of situations that meet these conditions. We also focus on finding the number of situations in two special cases: when exactly one seat is unoccupied whose total number is denoted by $ω(n)$, and when all audiences $1, \ldots, n-1$ are seated, whose total number is denoted by $ψ(n)$. Giving the recursive formula $ψ(n)=1+\sum_{d|n, d\neq n}ψ(d)$ with the initial value $ψ(1)=1$, we provide an explicit formula for $ψ(p^αq^β)$, where $p$ and $q$ are distinct prime numbers. Furthermore, we show that $ω(n)=-n+\sum_{i=1}^nψ(i)$.

math.GM↗
arXiv2026

Latin Eulerian Numbers

We introduce \emph{Latin Eulerian numbers} $\LE{n\atop k_1,\dots,k_n}$, a multivariate refinement of classical Eulerian numbers counting order-$n$ Latin squares by column ascents. We establish their fundamental symmetries, univariate marginals, and an exact multiplicity divisibility property. For the total ascent statistic $Σ(L)=\sum_i k_i(L)$, we prove the sharp, isolated bounds $n-1 \le Σ(L) \le (n-1)^2$, demonstrating that the adjacent values $n$ and $(n-1)^2-1$ are strictly unattainable. To analyze intermediate values, we show that symbol permutations typically ignored in classical enumeration directly govern $Σ$ via an exact formula. This insight yields explicit constructions for the interior range, motivates a unimodality conjecture for the total ascent distribution.

math.CO↗
arXiv2026

Prime and Touchard Congruences of Mixed-Type Bell Numbers

This paper establishes a comprehensive combinatorial and arithmetic framework for mixed Stirling and mixed Bell numbers, bridging partition structures, Touchard polynomials, and prime-power congruences. Furthermore, we develop to $p$-adic valuation theory, proving prime-power Touchard congruences and higher-order modulo-$p^2$ refinements that generalize classical arithmetic properties of combinatorial sequences.

math.CO↗
arXiv2026

Arithmetic Properties of Mixed Stirling Numbers of the second kind

We investigate the mixed Stirling numbers of the second kind, $\mathcal{S}(n; \mathbf{c})$, which count partitions of $n$ distinct elements into $m$ unlabeled and $k$ labeled non-empty blocks encoded by $\mathbf{c} = (m, 1^k)$. We establish their recurrence relations and exponential generating functions, and analyze their behavior modulo a prime $p$ and $p^2$. In particular, we extend the classical Touchard congruence to this mixed framework. Our results show that these configurations possess unique number-theoretic signatures distinct from classical set partitions.

math.CO↗
arXiv2026

On the Graphical $r$-Stirling Numbers of the First Kind for Specific Graph Families

This paper investigates the \textbf{graphical $r$-Stirling numbers of the first kind}, denoted by $\str{G}{k}$, which enumerate partitions of a vertex set $V(G)$ into $k$ disjoint cycles such that $r$ specified vertices occupy distinct blocks. We establish closed-form expressions and recursive identities for fundamental graph families, including \textbf{Path} ($P_n$), \textbf{Cycle} ($C_n$), \textbf{Star} ($S_n$), \textbf{Wheel} ($W_n$), and \textbf{Fan} ($F_n$) graphs. A primary focus of this study is the \textbf{statistical characterization} of the cycle distribution. We derive explicit formulas for the \textbf{mean} and \textbf{variance} of these numbers, extracted from the structural properties of the $r$-cycle polynomials. These results provide a rigorous measure of the average cycle density and variability across different graph topologies, bridging the gap between algebraic combinatorics and the structural analysis of restricted permutations.

math.CO↗
arXiv2020

Yet Another Generalization of The Notion of a Metric Space

A generalization of the triangle inequality is introduced by a mapping similar to a t-conorm mapping. This generalization leads us to a notion for which we use the $\star$-metric terminology. We are interested in the topological space induced by a $\star$-metric. Considering some examples of non-trivial $\star$-metrizable topological spaces, we also study the product topology for a finite family of $\star$-metrizable topological spaces.

math.GN↗
arXiv2019

A Characterization For 2-Self-Centered Graphs

A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing edge-minimal 2-self-centered graphs into two cases. First, we characterize edge-minimal 2-self-centered graphs without triangles by introducing \emph{specialized bi-independent covering (SBIC)} and a structure named \emph{generalized complete bipartite graph (GCBG)}. Then, we complete characterization by characterizing edge-minimal 2-self-centered graphs with some triangles. Hence, the main characterization is done since a graph is 2-self-centered if and only if it is a spanning subgraph of some edge-maximal 2-self-centered graphs and, at the same time, it is a spanning supergraph of some edge-minimal 2-self-centered graphs.

math.CO↗
arXiv2019

Mixed r-stirling numbers of the second kind

The Stirling number of the second kind $n\brace k$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls labeled $1$, $b_2$ balls labeled $2$, $\ldots$, $b_n$ balls labeled $n$ and $c_1+c_2+\ldots+c_k$ cells with $c_1$ cells labeled $1$, $c_2$ cells labeled $2$, $\ldots$, $c_k$ cells labeled $k$ and then we called the number of ways to partition the set of these balls into non-empty cells of these types as the \textit{mixed partition numbers}. As an application, we give a new statement of the $r$-Stirling numbers of second kind and $r$-Bell numbers. We also introduce the \textit{$r$-mixed Stirling number of second kind and $r$-mixed Bell numbers}. Finally, for a positive integer $m$ we evaluate the number of ways to write $m$ as the form $m_1\cdot m_2\cdot\ldots\cdot m_k$, where $k\geqslant 1$ and $m_i$'s are positive integers greater than $1$.

math.CO↗
arXiv2018

The Twelvefold way, the non-intersecting circles problem, and partitions of multisets

Let $n$ be a non-negative integer and $A=\{a_1,\ldots,a_k\}$ be a multi-set with $k$ not necessarily distinct members, where $a_1\leqslant\ldots\leqslant a_k$. We denote by $Δ(n,A)$ the number of ways to partition $n$ as the form $a_1x_1+\ldots+a_kx_k$, where $x_i$'s are distinct positive integers and $x_i< x_{i+1}$ whenever $a_i=a_{i+1}$. We give a recursive formula for $Δ(n,A)$ and some explicit formulas for some special cases. Using this notion we solve the non-intersecting circles problem which asks to evaluate the number of ways to draw $n$ non-intersecting circles in a plane regardless to their sizes. The latter also enumerates the number of unlabelled rooted tree with $n+1$ vertices.

math.CO↗
arXiv2018

Some results on ordered and unordered factorization of a positive integers

As a well-known enumerative problem, the number of solutions of the equation $m=m_1+...+m_k$ with $m_1\leqslant...\leqslant m_k$ in positive integers is $Π(m,k)=\sum_{i=0}^kΠ(m-k,i)$ and $Π$ is called the additive partition function. In this paper, we give a recursive formula for the so-called multiplicative partition function $μ_1(m,k):=$ the number of solutions of the equation $m=m_1... m_k$ with $m_1\leqslant...\leqslant m_k$ in positive integers. In particular, using an elementary proof, we give an explicit formula for the cases $k=1,2,3,4$.

math.CO↗
arXiv2016

Self-Contained Graphs

A self-contained graph is an infinite graph which is isomorphic to one of its proper induced subgraphs. In this paper, these graphs are studied by presenting some examples and defining some of their sub-structures such as removable subgraphs and the foundation. Then, we show that the general version of graph alternative conjecture, which says every graph has infinitely many strong twins or none, can be deduced from its connected version, which says every connected graph has infinitely many connected strong twins or none. Moreover, we try to find out under what conditions on two arbitrary removable subgraphs, their union is also a removable subgraph.

math.CO↗
arXiv2015

Strong Twins of Ordinary Star-Like Self-Contained Graphs

A self-contained graph is an infinite graph which is isomorphic to one of its proper induced subgraphs. In this paper, ordinary star-like self-contained graphs are introduced and it is shown that every ordinary star-like self-contained graph has infinitely many strong twins or none.

math.CO↗
arXiv2007

On minimal norms on $M_n$

In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on ${\mathbb C}^n$ such that $$N(A)=\max\{\|Ax\|_2: \|x\|_1=1, x\in {\mathbb C}^n\}$$ for all $A \in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.

math.FA↗
arXiv2006

Automatic Continuity of $σ$-Derivations on $C^*$-Algebras

Let $A$ be a $C^*$-algebra acting on a Hilbert space $H$, $σ:A\to B(H)$ be a linear mapping and $d:A\to B(H)$ be a $σ$-derivation. Generalizing the celebrated theorem of Sakai, we prove that if $σ$ is a continuous $*$-mapping then $d$ is automatically continuous. In addition, we show the converse is true in the sense that if $d$ is a continuous $*-σ$-derivation then there exists a continuous linear mapping $Σ:A\to B(H)$ such that $d$ is $*-Σ$-derivation. The continuity of the so-called $*-(σ,τ)$-derivations is also discussed.

math.FA↗