arXiv subjects

David G. L. Wang

Publications and source records attributed to David G. L. Wang.

At least 37 records · Page 2Linked to original sources

arXiv2020

The peak and descent statistics over ballot permutations

A ballot permutation is a permutation $π$ such that in any prefix of $π$ the descent number is not more than the ascent number. By using a reversal concatenation map, we give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, and connect this distribution and the joint distribution (pk, dp, des) of the peak, depth, and descent statistics over ordinary permutations in terms of generating functions. As corollaries, we obtain several formulas for the bivariate generating function for (i) the peak statistic over ballot permutations,(ii) the descent statistic over ballot permutations, and (iii) the depth statistic over ordinary permutations. In particular, we confirm Spiro's conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.

math.CO↗

arXiv2020

The edge metric dimension of the generalized Petersen graph $P(n,3)$ is 4

It is known that the problem of computing the edge dimension of a graph is NP-hard, and that the edge dimension of any generalized Petersen graph $P(n,k)$ is at least 3. We prove that the graph $P(n,3)$ has edge dimension 4 for $n\ge 11$, by showing semi-combinatorially the nonexistence of an edge resolving set of order 3 and by constructing explicitly an edge resolving set of order 4.

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

Geometry of limits of zeros of polynomial sequences of type $(1,2)$

We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of an arc, a circle, an interval or its exterior under the real line, the union of at most two different shapes of the above cases expect the union of an arc and a circle, and some degenerate forms.

math.CA↗

arXiv2020

A Toeplitz property of ballot permutations and odd order permutations

We give a new semi-combinatorial proof for the equality of the number of ballot permutations of length $n$ and the number of odd order permutations of length $n$, which is due to Bernardi, Duplantier and Nadeau. Spiro conjectures that the descent number of ballot permutations and certain cyclic weight of odd order permutations of the same length are equi-distributed. We present a bijection to establish a Toeplitz property for ballot permutations with any fixed number of descents, and a Toeplitz property for odd order permutations with any fixed cyclic weight. This allows us to refine Spiro's conjecture by tracking the neighbors of the largest letter in permutations.

math.CO↗

arXiv2020

Non-Schur-positivity of chromatic symmetric functions

We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we determine Schur positive fan graphs and Schur positive complete tripartite graphs. We show that any squid graph obtained by adding $n$ leaves to a common vertex on an $m$-vertex cycle is not Schur positive if $m\ne 2n-1$, and conjecture that neither are the squid graphs with $m=2n-1$.

math.CO↗

arXiv2018

The Tutte's condition in terms of graph factors

Let $G$ be a connected general graph of even order, with a function $f\colon V(G)\to\Z^+$. We obtain that $G$ satisfies the Tutte's condition \[ o(G-S)\le \sum_{v\in S}f(v)\qquad\text{for any nonempty set $S\subset V(G)$}, \] with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\colon V(G)\to 2^\N$ such that $H(v)\in \{J_f(v),\,J_f^+(v)\}$ for each $v\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a characterization for graphs of odd order satisfying the Tutte's condition with respect to a function.

math.CO↗

arXiv2018

Piecewise interlacing zeros of polynomials

We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly speaking, one constructs a collection of disjoint intervals such that one may show by induction that consecutive polynomials have interlacing zeros over each of the intervals. We confirm the real-rootedness of some polynomials satisfying a recurrence with linear polynomial coefficients. This extends Gross et al.'s work where one of the polynomial coefficients is a constant.

math.CO↗

arXiv2018

Geometry of limits of zeros of polynomial sequences of type $(1,1)$

In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, or a "lollipop", or an interval. As an application, we discover a sufficient and necessary condition for the universal real-rootedness of the polynomials, subject to certain sign condition on the coefficients of the recurrence. Moreover, we obtain the sharp bound for all the zeros when they are real.

math.CV↗

arXiv2018

Several variants of the Dumont differential system and permutation statistics

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (Math Comp, 1979, 33: 1293--1297) and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi-pairs of differential equations. We present a general method to derive the solutions of these differential equations. As applications, we present some characterizations for several permutation statistics.

math.CO↗

arXiv2017

Root geometry of polynomial sequences III: Type $(1,1)$ with positive coefficients

In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the overall real-rootedness of all the polynomials, in terms of the polynomial coefficients of the recurrence. Moreover, in the real-rooted case, we find the set of limits of zeros, which turns out to be the union of a closed interval and one or two isolated points; when non-real-rooted polynomial exists, we present a sufficient condition under which every polynomial with $n$ large has a real zero.

math.CO↗

arXiv2017

Common zeros of polynomials satisfying a recurrence of order two

We give a characterization of common zeros of a sequence of univariate polynomials $W_n(z)$ defined by a recurrence of order two with polynomial coefficients, and with $W_0(z)=1$. Real common zeros for such polynomials with real coefficients are studied further. This paper contributes to the study of root distribution of recursive polynomial sequences.

math.CO↗

arXiv2015

A Tutte-type characterization for graph factors

Let $G$ be a connected general graph. Let $f\colon V(G)\to \Z^+$ be a function. We show that $G$ satisfies the Tutte-type condition \[ o(G-S)\le f(S)\qquad\text{for all vertex subsets $S$}, \] if and only if it contains a colored $J_f^*$-factor for any $2$-end-coloring, where $J_f^*(v)$ is the union of all odd integers smaller than $f(v)$ and the integer $f(v)$ itself. This is a generalization of the $(1,f)$-odd factor characterization theorem, and answers a problem of Cui and Kano. We also derive an analogous characterization for graphs of odd orders, which addresses a problem of Akiyama and Kano.

math.CO↗

arXiv2015

The CLLC conjecture holds for cyclic outer permutations

Recently, Gross et al. posed the LLC conjecture for the locally log-concavity of the genus distribution of every graph, and provided an equivalent combinatorial version, the CLLC conjecture, on the log-concavity of the generating function counting cycles of some permutation compositions. In this paper, we confirm the CLLC conjecture for cyclic permutations, with the aid of Hultman numbers and by applying the Hermite--Biehler theorem on the generating function of Stirling numbers of the first kind. This leads to a further conjecture that every local genus polynomial is real-rooted.

math.CO↗

arXiv2015

The maximum number of perfect matchings of semi-regular graphs

Let $n\ge 34$ be an even integer, and $D_n=2\lceil n/4 \rceil-1$. In this paper, we prove that every $\{D_n,\,D_n+1\}$-graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings. This result is sharp in the sense that (i) there exists a $\{D_n,\,D_n+1\}$-graph containing exactly $\lceil n/4 \rceil$ disjoint perfect matchings, and that (ii) there exists a $\{D_n-1,\,D_n\}$-graph without perfect matchings for each $n$. As a consequence, for any integer $D\ge D_n$, every $\{D,\,D+1\}$-graph of order $n$ contains $\lceil (D+1)/2 \rceil$ disjoint perfect matchings. This extends Csaba et~al.'s breathe-taking result that every $D$-regular graph of sufficiently large order is $1$-factorizable, generalizes Zhang and Zhu's result that every $D_n$-regular graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings, and improves Hou's result that for all $k\ge n/2$, every $\{k,\,k+1\}$-graph of order $n$ contains $(\lfloor n/3\rfloor+1+k-n/2)$ disjoint perfect matchings.

math.CO↗

arXiv2015

Convolution Preserves Partial Synchronicity of Log-concave Sequences

In a recent proof of the log-concavity of genus polynomials of some families of graphs, Gross et al. defined the weakly synchronicity relation between log-concave sequences, and conjectured that the convolution operation by any log-concave sequence preserves weakly synchronicity. We disprove it by providing a counterexample. Furthermore, we find the so-called partial synchronicity relation between log-concave sequences, which is (i) weaker than the synchronicity, (ii) stronger than the weakly synchronicity, and (iii) preserved by the convolution operation.

math.CO↗

arXiv2015

Surface embedding of non-bipartite $k$-extendable graphs

We find the minimum number $k=μ'(Σ)$ for any surface $Σ$, such that every $Σ$-embeddable non-bipartite graph is not $k$-extendable. In particular, we construct the so-called bow-tie graphs $C_6\bowtie P_n$, and show that they are $3$-extendable. This confirms the existence of an infinite number of $3$-extendable non-bipartite graphs which can be embedded in the Klein bottle.

math.CO↗

arXiv2015

On colored set partitions of type $B_n$

Generalizing Reiner's notion of set partitions of type $B_n$, we define colored $B_n$-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored $B_n$-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored $B_n$-partition. We find an asymptotic expression of the total number of colored $B_n$-partitions up to an error of $O(n^{-1/2}\log^{7/2}{n})$, and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored $B_n$-partitions.

math.CO↗