arXiv subjects
Publications and source records attributed to Jiang Zeng.
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.
The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $γ$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.
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 $γ$-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 $q$-$γ$-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.
Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little Göllnitz identities. Noticing the connection between their results and Boulet's earlier four-parameter partition generating functions, we discover a new class of partitions, called $k$-strict partitions, to generalize their results. By applying both horizontal and vertical dissections of Ferrers' diagrams with appropriate labellings, we provide a unified combinatorial treatment of their results and shed more lights on the intriguing conditions of their companion to Capparelli's identities.
We show that certain terminating $_{6}ϕ_5$ series can be factorized into a product of two $_{3}ϕ_{2}$ series. As applications we prove a summation formula for a product of two $q$-Delannoy numbers along with some congruences for sums involving $q$-Delannoy numbers. This confirms three recent conjectures of the second author.
Two-dimensional transition metal dichalcogenides represent an emerging class of materials exhibiting various intriguing properties, and integration of such materials for potential device applications will necessarily encounter creation of different boundaries. Using first-principles approaches, here we investigate the structural, electronic, and magnetic properties along two inequivalent zigzag M and X edges of MX$_{2}$ (M=Mo, W; X=S, Se). Along the M edges, we reveal a previously unrecognized but energetically strongly preferred (2x1) reconstruction pattern, which is universally operative for all the MX$_{2}$, characterized by a self-passivation mechanism through place exchanges of the outmost X and M edge atoms. In contrast, the X edges undergo a more moderate (2x1) or (3x1) reconstruction for MoX$_{2}$ or WX$_{2}$, respectively. We further use the prototypical zigzag MoX$_{2}$ nanoribbons to demonstrate that the M and X edges possess distinctly different electronic and magnetic properties, which are discussed for spintronic and catalytic applications.
Lateral heterostructures of two-dimensional materials may exhibit various intriguing emergent properties. Yet when specified to the orientationally aligned heterojunctions of zigzag graphene and hexagonal boron nitride (hBN) nanoribbons, realizations of the high expectations on their properties encounter two standing hurtles. First, the rapid accumulation of strain energy prevents large- scale fabrication. Secondly, the pronounced half-metallicity predicted for freestanding graphene nanoribbons is severely suppressed. By properly tailoring orientational misalignment between zigzag graphene and chiral hBN nanoribbons, here we present a facile approach to overcome both obstacles. Our first-principles calculations show that the strain energy accumulation in such heterojunctions is significantly diminished for a range of misalignments. More strikingly, the half-metallicity is substantially enhanced from the orientationally aligned case, back to be comparable in magnitude with the freestanding case. The restored half-metallicity is largely attributed to the recovered superexchange interaction between the opposite heterojunction interfaces. The present findings may have important implications in eventual realization of graphene-based spintronics.
Carlitz has introduced q-analogues of the Bernoulli numbers around 1950. We obtain a representation of these q-Bernoulli numbers (and some shifted version) as moments of some orthogonal polynomials. This also gives factorisations of Hankel determinants of q-Bernoulli numbers, and continued fractions for their generating series. Some of these results are q-analogues of known results for Bernoulli numbers, but some are specific to the q-Bernoulli setting.
We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential operator having the $q$-classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the $q$-version of those given by Everitt {\it et al.} and by the first author, whilst the combinatorics of this new set of numbers is a $q$-version of the Jacobi-Stirling numbers given by Gelineau and the second author.
We consider several generalizations of the classical $γ$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $γ$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the literature.
We provide combinatorial interpretation for the $γ$-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding $γ$-coefficients for derangements. Our results refine the classical $γ$-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Brändén's modified Foata--Strehl action on permutations and the recent triple statistic (des, rix,aid) equidistibuted with (exc, fix, maj).
For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: $$ \sum_{k=0}^{\frac{p-1}{2}}{2k\choose k}^3\frac{1}{64^k} \equiv 0 \pmod{p^2} \quad\text{for}\quad p\equiv 3\pmod 4, $$ and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that \begin{align*} \sum_{k=0}^{\frac{p-1}{2}} {2k\brack k}_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0\pmod{[p]^2} \quad\text{for}\quad p\equiv 3\pmod 4, \\ \sum_{k=0}^{\frac{p-1}{2}}{2k\brack k}_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^2 } &\equiv 0 \pmod{[p]^2}\quad\text{for}\quad p\equiv 2\pmod{3}, \end{align*} where $[p]=1+q+\cdots+q^{p-1}$, $(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, and ${n\brack k}_q$ denotes the q-binomial coefficient. Actually, our results give q-analogues of Z.-H. Sun's and Z.-W. Sun's generalizations of the above Beukers-like supercongruences. Our proof uses the theory of basic hypergeometric series including a new q-Clausen-type summation formula.
We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv (-1)^{\frac{p-1}{2}}\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \sum_{k=0}^{p-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\varepsilon)k} &\equiv (-1)^{\frac{p-1}{2}}q^{\frac{(p^2-1)\varepsilon}{4}}\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \varepsilon=\pm1. Several related conjectures are also proposed.
We study a polynomial sequence $C_n(x|q)$ defined as a solution of a $q$-difference equation. This sequence, evaluated at $q$-integers, interpolates Carlitz-Riordan's $q$-ballot numbers. In the basis given by some kind of $q$-binomial coefficients, the coefficients are again some $q$-ballot numbers. We obtain in a combinatorial way another curious recurrence relation for these polynomials.
In analogy to the definition of the lambda-determinant, we define a one-parameter deformation of the Dodgson condensation formula for Pfaffians. We prove that the resulting rational function is a polynomial with weights given by the crossings and nestings of perfect matchings and prove several identities and closed-form evaluations.
Generalizing recent results of Egge and Mongelli, we show that each diagonal sequence of the Jacobi-Stirling numbers $\js(n,k;z)$ and $\JS(n,k;z)$ is a Pólya frequency sequence if and only if $z\in [-1, 1]$ and study the $z$-total positivity properties of these numbers. Moreover, the polynomial sequences $$\biggl\{\sum_{k=0}^n\JS(n,k;z)y^k\biggr\}_{n\geq 0}\quad \text{and} \quad \biggl\{\sum_{k=0}^n\js(n,k;z)y^k\biggr\}_{n\geq 0}$$ are proved to be strongly $\{z,y\}$-log-convex. In the same vein, we extend a recent result of Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising from the Lambert $W$ function, we obtain a neat proof of the unimodality of the latter sequence, which was proved previously by Kalugin and Jeffrey.
We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula.
Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.