arXiv subjects

Junyao Pan

Publications and source records attributed to Junyao Pan.

14 recordsLinked to original sources

arXiv2026

A note on the horizontal class transposition group

Let $n$ be an integer with $n > 1$. For every $r$ satisfying the inequalities $0 \leq r < n$, the residue class modulo $n$ is defined as $r(n)=\{r + kn | k \in Z\}$, where $Z$ is the set of all integers. Then for $0 \leq r_1\neq r_2 < n$, the horizontal class transposition $\tau_{r_1(n), r_2(n)}$ is an involution that interchanges $r_1 + kn$ and $r_2 + kn$ for each integer $k$ and fixes everything else. The horizontal class transposition group $CT_n$ is generated by all horizontal class transposition $\tau_{r_1(n), r_2(n)}$. Let $N$ be the least common multiple of the numbers $2, 3, . . . , n$ and $CT_{(n)}=\langle CT_2,CT_3,...,CT_n\rangle$. In this note, we prove that for $n>3$, $CT_{(n)}\cong S_N$, where $S_N$ is the symmetric group of degree $N$. Thus, we solve a conjecture proposed by Bardakov and Iskra, which has been included in the kourovka notebook: Unsolved problems in group theory, Novosibirsk, 2026.

math.GR↗

arXiv2025

Cuntz algebra automorphisms: transpositions

Permutative automorphisms of the Cuntz algebras $\mathcal{O}_n$ are in bijection with the stable permutations of $[n]^t$. They are also the elements of the reduced Weyl group of $Aut(\mathcal{O}_n)$. In this paper, we characterize the stability of transpositions in $S([n]^3)$, and thus providing a new family (with $6$ degrees of freedom) of automorphisms of the Cuntz algebras $\mathcal{O}_n$ for any $n>1$.

math.OA↗

arXiv2025

On two conjectures about pattern avoidance of cyclic permutations

Let $\pi$ be a cyclic permutation that can be expressed in its one-line form as $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and in its standard cycle form as $\pi = (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of pattern avoidance of one-line and the standard cycle form for a cyclic permutation $\pi$, defined as both $\pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and its standard cycle form $c_1c_2\cdot\cdot\cdot c_{n}$ avoiding a given pattern. Let $\mathcal{A}_n(\sigma_1,...,\sigma_k; \tau)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid each pattern of $\{\sigma_1,...,\sigma_k\}$ in their one-line forms and avoid $\tau$ in their standard cycle forms. In this paper, we obtain some results about the cyclic permutations avoiding patterns in both one-line and cycle forms. In particular, we resolve two conjectures of Archer et al.

math.CO↗

arXiv2025

On finite groups whose coprime graph is a divisor graph

In this paper, we first characterize which generalized lexicographic products are divisor graphs. As applications, we show that power graphs, reduced power graphs and order graphs are all divisor graphs, which also implies the main result in [Power graph of a finite group is always divisor graph, Asian-European Journal of Mathematics 16 (2023)]. We then show that, the coprime graph of a group is a generalized lexicographic product, and characterize which coprime graphs are divisor graphs. Finally, we classify the finite groups $G$ having at most four prime divisors, whose coprime graphs are divisor graphs, and we also classify the finite groups $G$ whose coprime graphs are divisor graphs, if $G$ is a nilpotent group, a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, a direct product of two non-trivial groups, and a sporadic simple group.

math.GR↗

arXiv2024

A note on the Cuntz algebra automorphisms

Permutative automorphisms of the Cuntz algebras $\mathcal{O}_n$ are in bijection with the stable permutations of $[n]^k$. They are also the elements of the restricted Weyl group of $Aut(\mathcal{O}_n)$. In this note, we characterize a class of stable involutions of $[n]^2$. More precisely, we prove Conjecture 12.2 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], and thus providing a new family (with $6$ degrees of freedom) of automorphisms of the Cuntz algebras $\mathcal{O}_n$ for any $n>1$.

math.GR↗

arXiv2024

On a conjecture about pattern avoidance of cycle permutations

Let $\pi$ be a cycle permutation that can be expressed as one-line $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and a cycle form $\pi = (c_1,c_2, ..., c_n)$. Archer et al. introduced the notion of pattern avoidance of one-line and all cycle forms for a cycle permutation $\pi$, defined as $\pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and its arbitrary cycle form $c_ic_{i+1}\cdot\cdot\cdot c_nc_1c_2\cdot\cdot\cdot c_{i-1}$ avoid a given pattern. Let $\mathcal{A}^\circ_n(\sigma; \tau)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid $\sigma$ in their one-line form and avoid $\tau$ in their all cycle forms. In this note, we prove that $|\mathcal{A}^\circ_n(2431; 1324)|$ is the $(n-1)^{\rm{st}}$ Pell number for any positive integer $n$. Thereby, we give a positive answer to a conjecture of Archer et al.

math.CO↗

arXiv2024

Automorphism group of the graph $A(n,k,r)$

Let $[n]^{(k)}$ be the set of all ordered $k$-tuples of distinct elements in $[n]=\{1,2,...,n\}$. The $(n,k,r)$-arrangement graph $A(n,k,r)$ with $1\leq r\leq k\leq n$, is the graph with vertex set $[n]^{(k)}$ and with two $k$-tuples are adjacent if they differ in exactly $r$ coordinates. In this manuscript, we characterize the full automorphism groups of $A(n,k,r)$ in the cases that $1\leq r=k\leq n$ and $r=2<k=n$. Thus, we resolve two special cases of an open problem proposed by Fu-Gang Yin, Yan-Quan Feng, Jin-Xin Zhou and Yu-Hong Guo. In addition, we conclude with a bold conjecture.

math.GR↗

arXiv2024

On the permutations that strongly avoid the pattern 312 or 231

In 2019, B\'ona and Smith introduced the notion of \emph{strong pattern avoidance}, that is, a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations $\pi$ which not only strongly avoid the pattern $312$ or $231$ but also avoid the pattern $\tau$, for $\tau\in S_3$ and some $\tau\in S_4$. One of them is to give a positive answer to a conjecture of Archer and Geary.

math.CO↗

arXiv2023

A Note On The Cross-Sperner Families

Let $(\mathcal{F},\mathcal{G})$ be a pair of families of $[n]$, where $[n]=\{1,2,...,n\}$. If $A\not\subset B$ and $B\not\subset A$ hold for all $A\in\mathcal{F}$ and $B\in\mathcal{G}$, then $(\mathcal{F},\mathcal{G})$ is called a Cross-Sperner pair. P. Frankl and Jian Wang introduced the extremal problem that $m(n)={\rm{max}}\{|\mathcal{I}(\mathcal{F},\mathcal{G})|:\mathcal{F},\mathcal{G}\subset2^{[n]}~{\rm{are~cross}}$-${\rm{sperner}}\}$, where $\mathcal{I}(\mathcal{F},\mathcal{G})=\{A\cap B:A\in\mathcal{F},B\in\mathcal{G}\}$. In this note, we prove that $m(n)=2^n-2^{\lfloor\frac{n}{2}\rfloor}-2^{\lceil\frac{n}{2}\rceil}+1$ for all $n>1$. This solves an open problem proposed by P. Frankl and Jian Wang.

math.CO↗

arXiv2023

The full automorphism groups of general position graphs

Let $S$ be a non-empty finite set. A flag of $S$ is a set $f$ of non-empty proper subsets of $S$ such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. The set $\{|X|:X\in f\}$ is called the type of $f$. Two flags $f$ and $f'$ are in general position with respect to $S$ if $X\cap Y=\emptyset$ or $X\cup Y=S$ for all $X\in f$ and $Y\in f'$. For a fixed type $T$, Klaus Metsch defined the general position graph $\Gamma(S,T)$ whose vertices are the flags of $S$ of type $T$ with two vertices being adjacent when the corresponding flags are in general position. In this paper, we characterize the full automorphism groups of $\Gamma(S,T)$ in the case that $|T|=2$. In particular, we solve an open problem proposed by Klaus Metsch.

math.GR↗

arXiv2023

Sign-Balanced Pattern-Avoiding Permutation Classes

A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations. Let $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ be the set of permutations in the symmetric group $S_n$ which avoids patterns $\sigma_1, \sigma_2, \ldots, \sigma_r$. The aim of this paper is to investigate when, for certain patterns $\sigma_1, \sigma_2, \ldots, \sigma_r$, $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ is sign-balanced for every integer $n>1$. We prove that for any $\{\sigma_1, \sigma_2, \ldots, \sigma_r\}\subseteq S_3$, if $\{\sigma_1, \sigma_2, \ldots, \sigma_r\}$ is sign-balanced except $\{132, 213, 231, 312\}$, then $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ is sign-balanced for every integer $n>1$. In addition, we give some results in the case of avoiding some patterns of length $4$.

math.CO↗

arXiv2021

Two Identities

In the process of studying a conjecture of Holly M. Green and Martin W. Liebeck, we obtain two interesting identities by elementary methods, one is a combinatorial identity, and the other is a number theoretic identity.

math.GM↗

arXiv2020

The generating pairs of the 2-transitive groups

Given a finite group $G$. The generating pair $(H,a)$ of $G$, that is, $H<G$ and $a\in G$ such that $\langle a,H\rangle=G$. In this paper, we introduce the definition of FF-subgroup to characterize the generating pairs of the symmetric groups, alternating groups and projective groups $PSL(2,q)$. This gives a partial answer to an open problem of J. Andr\'e and J. Ara\'ujo and P. J. Cameron.

math.GR↗

arXiv2018

On simultaneous conjugation of permutations

In this paper we give an affirmative answer to a conjecture proposed by Danny Neftin, that is, if the commutator of two permutations has at least n-4 fixed points where two permutations are in degree n symmetric group, then there exists a permutation in this symmetric inverting both of them.

math.GR↗