arXiv subjects

Yueming Zhong

Publications and source records attributed to Yueming Zhong.

9 recordsLinked to original sources

arXiv2026

Transfer Matrices and Ehrhart Theory for Path and Cyclic Block Polytopes

We study block polytopes whose variables are divided into equal-size blocks and whose local inequalities bound the total contribution of adjacent blocks. For blocks arranged along a path, we develop a transfer-matrix enumeration in the length direction. We also carry out an Ehrhart-theoretic analysis in the dilation direction. The original transfer matrix admits a compression to a weighted height matrix, and the numerator and denominator of the length generating function are described by explicit recurrences and determinant formulas. We also study the cyclic analogue, where the length generating function is governed by the logarithmic derivative of the same determinant. On the Ehrhart side, the path polytopes with at least two blocks and the even cyclic polytopes are stable-set polytopes of perfect graphs; consequently they are Gorenstein of codegree $2a + 1$ (independent of the number of blocks $m \ge 2$), satisfy an explicit Ehrhart--Macdonald reciprocity, and have palindromic unimodal $h^*$-polynomials of degree $a(m-2)$. Odd cyclic polytopes have denominator exactly two, and their lattice-point enumerators are Ehrhart quasipolynomials of period dividing two. Further combinatorial interpretations and open problems are discussed.

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arXiv2026

Alternating adjacent-sum polytopes: transfer matrices and Ehrhart series

We study a period-two family of adjacent-sum lattice polytopes whose consecutive-coordinate bounds alternate between $s$ and $s+1$. This provides a simple non-uniform deformation of the classical uniform model while retaining an explicit transfer-matrix structure. The lattice-point counts exhibit a parity split: the odd- and even-dimensional sequences have distinct rational generating functions with a common denominator. The odd-dimensional series satisfies a M\"obius recurrence and admits an arctangent closed form, whereas the even-dimensional series obeys a coupled recurrence. Their common dominant pole determines the exponential growth in both parity classes. For the cyclic model obtained by adding a constraint between the first and last coordinates, the count becomes a matrix trace. The two cyclic parity classes again have rational generating functions with the same denominator; the even-dimensional numerator has a Jacobi-derivative form, while the odd-dimensional one is given by an explicit anti-diagonal cofactor expression. We also derive dimension-generating functions for fixed dilations, linear recurrences for lattice-point counts, rational volume-generating functions, and a bivariate identity for the coefficients of the $h^*$-polynomials. When $s=1$, every even-dimensional polytope decomposes into a Cartesian product of unimodular triangles, yielding explicit formulas and the Gorenstein property. For every $s\geq 2$, the Gorenstein property fails in some even dimension.

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arXiv2026

Magic labelling enumeration on pseudo-line graphs and pseudo-cycle graphs

Stanley's theorem establishes that for any finite graph $G$, the number $h_G(s)$ of magic labelings with magic sum $s$ can be expressed as a sum of two polynomials in $s$. However, determining the precise form of $h_G(s)$ is generally challenging. This paper aims to compute $h_G(s)$ and its generating function for pseudo-line graphs and pseudo-cycle graphs, thereby extending the earlier work of B\'{o}na et al.\cite{Bona-1,Bona}.

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arXiv2022

The constant term algebra of type $A$: the Structure

In this paper, we discover a new noncommutative algebra. We refer this algebra as the constant term algebra of type $A$, which is generated by certain constant term operators. We characterize a structural result of this algebra by establishing an explicit basis in terms of certain forests. This algebra arises when we apply the method of the iterated Laurent series to investigate Beck and Pixton's residue computation for the Ehrhart series of the Birkhoff polytope. This algebra seems to be the first structural result in the area of the constant term world since the discovery of the Dyson constant term identity in 1962.

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arXiv2022

Proving some conjectures on Kekul\'{e} numbers for certain benzenoids by using Chebyshev polynomials

In chemistry, Cyvin-Gutman enumerates Kekul\'{e} numbers for certain benzenoids and record it as $A050446$ on OEIS. This number is exactly the two variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where $T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to both the row and column generating function of $T(n,m)$. In this paper, give explicit formula of the column generating function, which is also the generating function $F(n,x)$ studied by B\'{o}na, Ju, and Yoshida. We also get trig function representations by using Chebyshev polynomials of the second kind. This allows us to prove all these conjectures.

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arXiv2021

A symmetric chain decomposition of $N(m,n)$ of composition

A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $\alpha=(\alpha_1,\cdots,\alpha_m)$, with $0\le \alpha_i \le n$ for each $i=1,\cdots,m$. Define order $<$ as follow, $\forall \alpha,\beta \in N(m,n)$, $\beta < \alpha$ if and only if $\beta_i \le \alpha_i(i=1,\cdots,m)$ and $\sum\limits_{i=1}^{m}\beta_i <\sum\limits_{i=1}^{m}\alpha_i$. In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method.

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arXiv2021

On Magic Distinct Labellings of Simple Graphs

A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave remarkable results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph $G$. The idea is illustrated in detail by dealing with three regular graphs. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct labellings.

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arXiv2021

Constructing explicit Sperner chain decompositions for $L(3,n)$ and $L(4,n)$ via Greedy Algorithms and chain tableaux

Let $L(m,n)$ denote Young's lattice, consisting of all partitions whose Young diagrams are contained within an $m\times n$ rectangle. It is a classical result that the partially ordered set $L(m,n)$ is rank-symmetric, rank-unimodal, and Sperner; however, finding a direct combinatorial proof via an explicit order matching remains a prominent open problem in the field. In this paper, we address this challenge by constructing explicit order matchings for $L(3,n)$ and extending our methods to comprehensively cover $L(4,n)$. To achieve this, we introduce a novel ``chain tableau" representation, which serves as a powerful tool for identifying and characterizing complex combinatorial patterns. Notably, we demonstrate that the same order matchings can be independently derived using both a greedy algorithm and a recursive kneading process. This work not only resolves the explicit matching problem for $m=3$ and $m=4$ but also establishes robust structural tools that may offer valuable insights into the general $L(m,n)$ case.

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arXiv2019

On Parity Unimodality of $q$-Catalan Polynomials

A polynomial $A(q)=\sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0\le a_1\le \cdots \le a_k\ge a_{k+1} \ge \cdots \ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= \frac{1}{[n+m]} \left[ m+n \atop n\right]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $m\le 5$ in a straightforward way.

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