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In 1977 Carlitz and Scoville introduced the cycle $(\alpha,t)$-Eulerian polynomials $A^{\mathrm{cyc}}_n(x,y, t\,|\,\alpha)$ by enumerating permutations with respect to the number of excedances, drops, fixed points and cycles. In this paper, we introduce a nine-variable generalization of the Eulerian polynomials $A_n(u_1,u_2,u_3,u_4, f, g, t\,|\,\alpha, \beta)$ in terms of descent based statistics of permutations and prove a connection formula between these two kinds of generalized Eulerian polynomials. By exploring the connection formula, we derive plainly the exponential generating function of the latter polynomials and various $\gamma$-positive formulas for variants of Eulerian polynomials. In particular, our results unify and strengthen the recent results by Ji and Ji-Lin. In related work to the transition matrix between the Specht and web bases, Hwang, Jang and Oh recently introduced the web permutations, which can be characterised by cycle Andr\'e permutations. We show that enumerating the latter permutations with respect to the number of drops, fixed points and cycles gives rise to the normalised $\gamma$-vectors of the $(\alpha,t)$-Eulerian polynomials. Our result generalizes and unifies several known results in the literature., Comment: 37 pages

Recently Cheng et al. (Adv. in Appl. Math. 143 (2023) 102451) generalized the inversion number to partial permutations, which are also known as Laguerre digraphs, and asked for a suitable analogue of MacMahon's major index. We provide such a major index, namely, the corresponding maj and inv statistics are equidistributed, and exhibit a Haglund-Remmel-Wilson type identity. We then interpret some Jacobi-Rogers polynomials in terms of Laguerre digraphs generalizing Deb and Sokal's alternating Laguerre digraph interpretation of some special Jacobi-Rogers polynomials.

Materials with ferroelectrically switchable topological properties are of interest for both fundamental physics and practical applications. Using first-principles calculations, we find that stacking ferroelectric $\alpha$-In$_2$Se$_3$ monolayers into a bilayer leads to polarization-dependent band structures, which yields polarization-dependent topological properties. Specifically, we find that the states with interlayer ferroelectric couplings are quantum spin Hall insulators, while those with antiferroelectric polarizations are normal insulators. We further find that In$_2$Se$_3$ trilayer and quadlayer exhibit nontrivial band topology as long as in the structure the ferroelectric In$_2$Se$_3$ bilayer is antiferroelectrically coupled to In$_2$Se$_3$ monolayers or other ferroelectric In$_2$Se$_3$ bilayer. Otherwise the system is topologically trivial. The reason is that near the Fermi level the band structure of the ferroelectric In$_2$Se$_3$ bilayer has to be maintained for the nontrivial band topology. This feature can be used to design nontrivial band topology for the thicker films by a proper combination of the interlayer polarization couplings. The topological properties can be ferroelectrically tunable using the dipole locking effect. Our study reveals switchable band topology in a family of natural ferroelectrics, which provide a platform for designing new functional devices., Comment: 12 pages, 12 figures

We prove two recent conjectures of Bourn and Erickson (2023) regarding the real-rootedness of a certain family of polynomials $N_n(t)$ as well as the sum of their coefficients. These polynomials arise as the numerators of generating functions in the context of the discrete one-dimensional earth mover's distance (EMD) and have also connection to the Wiener index of minuscule lattices. We also prove that the coefficients of $N_n(x)$ are asymptotically normal, the coefficient matrix of $N_n(x)$ is totally positive and the polynomial sequence $N_n(x)$'s is $x$-log-concave., Comment: 13 pages, some typos are corrected

The recent interest in $q$-Stirling numbers of the second kind in type B prompted us to give a type B analogue of a classical identity connecting the $q$-Stirling numbers of the second kind and Carlitz's major $q$-Eulerian numbers, which turns out to be a $q$-analogue of an identity due to Bagno, Biagioli and Garber. We provide a combinatorial proof of this identity and an analytical proof of a more general identity for colored permutations. In addition, we prove some $q$-identities about the $q$-Stirling numbers of the second kind in types A, B and D., Comment: 24 pages, to appear in Electron. J. Combin

We prove a recent conjecture, due to Vigren and Dieckmann, about an explicit triple sum formula for a series from Ramanujan's Notebooks. We shall give two proofs: the first one is by evaluation and based on the identity \begin{equation*} \sum_{k=0}^\infty \frac{(x+k)^{m+k}}{k!}e^{-u(x+k)} u^k = \sum_{j=0}^\infty \sum_{i=0}^{m}\binom{m+j}{i} \stirl{m+j-i}{j}x^iu^j, \end{equation*} where $\genfrac\{\}{0pt}{}{n}{k}$ is a Stirling number of the second kind, and the second one is combinatorial in nature and by induction., Comment: 7 pages

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., Comment: 28 pages, final version, accepted for publication in Discrete Mathematics

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B., Comment: Advances in Applied Mathematics 152, 102599

Strong band engineering in two-dimensional (2D) materials can be achieved by introducing moir\'e superlattices, leading to the emergence of various novel quantum phases with promising potential for future applications. Presented works to create moir\'e 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{\deg} 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 moir\'e 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., Comment: 13 pages, 4 figures

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.

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 $\gamma$-coefficients of our $(p,q)$-analogue of descent polynomials have the same factorization flavor as the $\gamma$-coeffcients of Br\"and\'en's $(p,q)$-Eulerian polynomials., Comment: 30 pages, 5 figures, typos fixed

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., Comment: 24 pages, typos fixed, accepted for publication in The Ramanujan Journal

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., Comment: 25 pages, 5 figures, revised based on referees report

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., Comment: 8 pages

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.

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 moir\'{e} flat bands, and hence opens new opportunities to study the valleytronics of mori\'{e} 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.

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., Comment: 15 pages,3 figures, comments are welcome

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 Andr\'e permutations} and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\em signed Andr\'e permutations} and {\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$., Comment: 21 pages, 3 figures, 6 tables

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., Comment: 12 pages

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them., Comment: LaTeX2e, 122 pages, includes 9 tikz figures. Version 2 includes a new footnote 27 on page 81. To be published in Advances in Applied Mathematics

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., Comment: 5 pages, 4 figures

We construct bijections to show that two pairs of sextuple set-valued statistics of permutations are equidistributed on symmetric groups. This extends a recent result of Sokal and the second author valid for integer-valued statistics as well as a previous result of Foata and Han for bivariable set-valued statistics., Comment: 14 pages

In 2008 Br\"and\'en proved a $(p,q)$-analogue of the $\gamma$-expansion formula for Eulerian polynomials and conjectured the divisibility of the $\gamma$-coefficient $\gamma_{n,k}(p,q)$ by $(p+q)^k$. As a follow-up, in 2012 Shin and Zeng showed that the fraction $\gamma_{n,k}(p, q)/(p + q)^k$ 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 Andr\'e permutations, a class of objects first defined and studied by Foata, Sch\"utzenberger 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.

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 $\gamma$-coefficients of the inversion polynomials restricted on $321$-avoiding permutations., Comment: 40 pages, 5 figures. To appear in Advances in Applied Mathematics

We consider a $(q,y)$-analogue of Laguerre polynomials $L^{(\alpha)}_n(x;y;q)$ for integral $\alpha\geq -1$, which turns out to be a rescaled version of Al-Salam--Chihara polynomials. A combinatorial interpretation for the $(q,y)$-Laguerre polynomials is given using a colored version of Foata-Strehl's Laguerre configurations with suitable statistics. When $\alpha\geq 0$, the corresponding moments are described using certain classical statistics on permutations, and the linearization coefficients are proved to be a polynomial in $y$ and $q$ with nonnegative integral coefficients., Comment: The typo for the signless (q,y)-Laguerre polynomials is fixed in (1.8) and subsequently

We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$\gamma$-positivity using group actions. Motivated by the sign-balance identity of D\'esarm\'enien--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 $\gamma$-positivity of binomial-Eulerian polynomials which involves crossings and nestings of permutations., Comment: 19 pages (without Sections 5,6 of the previous version)

The aim of this paper is two-fold. We first prove several new interpretations of a kind of $(q,t)$-Catalan numbers along with their corresponding $\gamma$-expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain $(-1)$-phenomenon for each subset of permutations avoiding a single pattern of length three, and discuss their $q$-analogues utilizing the newly obtained $q$-$\gamma$-expansions, as well as the continued fraction of a quint-variate generating function due to Shin and the fourth author. Moreover, we enumerate the alternating permutations avoiding simultaneously two patterns, namely $(2413,3142)$ and $(1342,2431)$, of length four, and consider such $(-1)$-phenomenon for these two subsets as well., Comment: 36 pages, 3 figures, 4 tables