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Jiang Zeng

Publications and source records attributed to Jiang Zeng.

At least 37 records · Page 2Linked to original sources
arXiv2023

Enumeration of permutations by the parity of descent positions

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent positions, we prove a $q$-analogue of Carlitz-Scoville's generating function by counting the inversion number and a type B analogue by enumerating the signed permutations with respect to the parity of desecnt and ascent positions. As a by-product of our formulas, we obtain a $q$-analogue of Chebikin's formula for alternating descent polynomials, an alternative proof of Sun's gamma-positivity of her bivariate Eulerian polynomials and a type B analogue, the latter refines Petersen's gamma-positivity of the type B Eulerian polynomials.

math.CO↗
arXiv2022

Transport evidence of superlattice Dirac cones in graphene monolayer on twisted boron nitride substrate

Strong band engineering in two-dimensional (2D) materials can be achieved by introducing moiré superlattices, leading to the emergence of various novel quantum phases with promising potential for future applications. Presented works to create moiré patterns have been focused on a twist embedded inside channel materials or between channel and substrate. However, the effects of a twist inside the substrate materials on the unaligned channel materials are much less explored. In this work, we report the realization of superlattice multi-Dirac cones with the coexistence of the main Dirac cone in a monolayer graphene (MLG) on a ~0.14° twisted double-layer boron nitride (tBN) substrate. Transport measurements reveal the emergence of three pairs of superlattice Dirac points around the pristine Dirac cone, featuring multiple metallic or insulating states surrounding the charge neutrality point (CNP). Displacement field tunable and electron-hole asymmetric Fermi velocities are indicated from temperature dependent measurements, along with the gapless dispersion of superlattice Dirac cones. The experimental observation of multiple Dirac cones in MLG/tBN heterostructure is supported by band structure calculations employing periodic moiré potential. Our results unveil the potential of using twisted substrate as a universal band engineering technique for 2D materials regardless of lattice matching and crystal orientations, which might pave the way for a new branch of twistronics.

cond-mat.mes-hall↗
arXiv2022

Minor summation formula of hyperpfaffians and Selberg integrals

In the previous paper (J. Combin. Theory Ser. A, 120, 2013, 1263--1284) H. Tagawa and the two authors proposed an algebraic method to compute certain Pfaffians whose form resemble to Hankel determinants associated with moment sequences of the classical orthogonal polynomials. At the end of the paper they offered several conjectures. In this work we employ a completely different method to evaluate this type of Pfaffians. The idea is to apply certain de Bruijn type formula and convert the evaluation of the Pfaffians to the certain Selberg type integrals. This approach works not only for Pfaffians but also for hyperpfaffians. Hence it enables us to establish much more generalized identities than those conjectured in the previous paper. We also investigate some Pfaffians related to classical $q$-orthogonal polynomials.

math.CO↗
arXiv2022

Moments of Orthogonal Polynomials and Exponential Generating Functions

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont-Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally we show that one of our methods can be applied to deal with the moments of Askey-Wilson polynomials.

math.CA↗
arXiv2021

Equidistributions around special kinds of descents and excedances

We consider a sequence of four variable polynomials by refining Stieltjes' continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections we derive various combinatorial interpretations in terms of permutation statistics for these polynomials, which include special kinds of descents and excedances in a recent paper of Baril and Kirgizov. As a by-product, we derive several equidistribution results for permutation statistics, which enables us to confirm and strengthen a recent conjecture of Vajnovszki and also to obtain several compagnion permutation statistics for two bistatistics in a conjecture of Baril and Kirgizov.

math.CO↗
arXiv2021

Multi-orbital model reveals second-order topological insulator in 1H-transition metal dichalcogenide

Recently, a new class of second-order topological insulators (SOTIs) characterized by an electronic dipole has been theoretically introduced and proposed to host topological corner states. As a novel topological state, it has been attracting great interest and experimentally realized in artificial systems of various fields of physics based on multi-sublattice models, e.g., breathing kagome lattice. In order to realize such kind of SOTI in natural materials, we proposed a symmetry-faithful multi-orbital model. Then, we reveal several familiar transition metal dichalcogenide (TMD) monolayers as a material family of two-dimensional SOTI with large bulk gaps. The topologically protected corner state with fractional charge is pinned at Fermi level due to the charge neutrality and filling anomaly. Additionally, we propose that the zero-energy corner state preserves in the heterostructure composed of a topological nontrivial flake embedded in a trivial material. The novel second-order corner states in familiar TMD materials hold promise for revealing unexpected quantum properties and applications.

cond-mat.mes-hall↗
arXiv2021

Cycles of even-odd drop permutations and continued fractions of Genocchi numbers

Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and E-permutations) for (median) Genocchi numbers. They further conjecture that the distributions of cycle numbers over the two models are equal. In a follow-up, Eu et al. (arXiv:2103.09130) further proved the gamma-positivity of the descent polynomials of even-odd descent permutations, which are in bijection with E-permutations by Foata's fundamental transformation. This paper merges the above two papers by considering a general moment sequence which encompasses the number of cycles and number of drops of E-permutations. Using the combinatorial theory of continued fraction, the moment connection enables us to confirm Lazar-Wachs' conjecture and obtain a natural $(p,q)$-analogue of Eu et al's descent polynomials. Furthermore, we show that the $γ$-coefficients of our $(p,q)$-analogue of descent polynomials have the same factorization flavor as the $γ$-coeffcients of Brändén's $(p,q)$-Eulerian polynomials.

math.CO↗
arXiv2021

Valley-selective Floquet Chern Flat Bands in Twisted Multilayer Graphene

We show that Floquet engineering with circularly polarized light (CPL) can selectively split the valley degeneracy of a twisted multilayer graphene (TMG), and thus generate a controlled valley-polarized Floquet Chern flat band with tunable large Chern number. It offers a feasible optical way to manipulate the valley degree of freedom in moiré flat bands, and hence opens new opportunities to study the valleytronics of morié flat band systems. We thus expect that many of the valley-related properties of TMG, e.g. orbital ferromagnetism, can be switched by CPL with proper doping. We reveal a Chern number hierarchy rule for the Floquet flat bands in a generic (M+N)-layer TMG. We also illustrate that the CPL effects on TMG strongly rely on the stacking chirality, which is an unique feature of TMG. All these phenomena could be tested in the twisted double bilayer graphene systems, which is the simplest example of TMG and has already been realized in experiment.

cond-mat.mes-hall↗
arXiv2021

Realistic flat-band model based on degenerate $p$-orbitals in two-dimensional ionic materials

Though several theoretical models have been proposed to design electronic flat-bands, the definite experimental realization in two-dimensional atomic crystal is still lacking. Here we propose a novel and realistic flat-band model based on threefold degenerate $p$-orbitals in two-dimensional ionic materials. Our theoretical analysis and first-principles calculations show that the proposed flat-band can be realized in 1T layered materials of alkali-metal chalogenides and metal-carbon group compounds. Some of the former are theoretically predicted to be stable as layered materials (e.g., K$_2$S), and some of the latter have been experimentally fabricated in previous works (e.g., Gd$_2$CCl$_2$). More interestingly, the flat-band is partially filled in the heterostructure of a K$_2$S monolayer and graphene layers. The spin polarized nearly flat-band can be realized in the ferromagnetic state of a Gd$_2$CCl$_2$ monolayer, which has been fabricated in experiments. Our theoretical model together with the material predictions provide a realistic platform for the study of flat-bands and related exotic quantum phases.

cond-mat.mtrl-sci↗
arXiv2020

Moments of q-Jacobi Polynomials and q-Zeta Values

We explore some connections between moments of rescaled little q-Jacobi polynomials, q-analogues of values at negative integers for some Dirichlet series, and the q-Eulerian polynomials of wreath products of symmetric groups.

math.CO↗
arXiv2020

More bijections for Entringer and Arnold families

The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$ alternating permutations (resp. starting with $k$). In this paper, using bijections we first derive the counterparts in {\em André permutations} and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\em signed André permutations} and {\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$.

math.CO↗
arXiv2020

Basic and bibasic identities related to divisor functions

Using basic hypergeometric functions and partial fraction decomposition we give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hamme, Prodinger, and Chen-Fu related to divisor functions. An identity relating Lambert series to Eulerian polynomials is also proved.

math.CO↗
arXiv2020

Eulerian polynomials and excedance statistics

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulae are comparable with Zhuang's generalizations [Adv. in Appl. Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng [European J. Combin. 33 (2012), no. 2, 111--127] and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the $γ$-coefficients of the inversion polynomials restricted on $321$-avoiding permutations.

math.CO↗
arXiv2020

Equidistributions of mesh patterns of length two and Kitaev and Zhang's conjectures

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more equidistributions of mesh pattern and confirm Kitaev and Zhang's four conjectures by constructing two involutions on permutations.

math.CO↗
arXiv2020

Around the $q$-binomial-Eulerian polynomials

We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$γ$-positivity using group actions. Motivated by the sign-balance identity of Désarménien--Foata--Loday for the $(\mathrm{des}, \mathrm{inv})$-Eulerian polynomials, we further investigate the sign-balance of the $q$-binomial-Eulerian polynomials. We show the unimodality of the resulting signed binomial-Eulerian polynomials by exploiting their continued fraction expansion and making use of a new quadratic recursion for the $q$-binomial-Eulerian polynomials. We finally use the method of continued fractions to derive a new $(p,q)$-extension of the $γ$-positivity of binomial-Eulerian polynomials which involves crossings and nestings of permutations.

math.CO↗
arXiv2020

Gap between the largest and smallest parts of partitions and Berkovich and Uncu's conjectures

We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263--284) on the inequalities between the numbers of partitions of $n$ with bounded gap between largest and smallest parts for sufficiently large $n$. Actually our theorems are stronger than their original conjectures. The analytic version of our results shows that the coefficients of some partition $q$-series are eventually positive.

math.CO↗
arXiv2020

Brändén's $(p,q)$-Eulerian polynomials, André permutations and continued fractions

In 2008 Brändén proved a $(p,q)$-analogue of the $γ$-expansion formula for Eulerian polynomials and conjectured the divisibility of the $γ$-coefficient $γ_{n,k}(p,q)$ by $(p+q)^k$. As a follow-up, in 2012 Shin and Zeng showed that the fraction $γ_{n,k}(p, q)/(p + q)^k$ is a polynomial in $\N[p,q]$. The aim of this paper is to give a combinatorial interpretation of the latter polynomial in terms of André permutations, a class of objects first defined and studied by Foata, Schützenberger and Strehl in the 1970s. It turns out that our result provides an answer to a recent open problem of Han, which was the impetus of this paper.

math.CO↗