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Yingrui Zhang

Publications and source records attributed to Yingrui Zhang.

15 recordsLinked to original sources
arXiv2026

Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths

Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are $q,t$-symmetric by constructing an involution on plane trees. They also provided a new formula for the original $q,t$-Catalan polynomials $C_{n}(q,t)$. We observe that depth is a slight modification of bounce, which was defined by the filling algorithm and ranking algorithm of Xin and the second author in their study of $\vec{k}$-Dyck paths. In this article, we generalize depth of classical Dyck paths to the case of $\vec{k}$-Dyck paths and prove $q,t$-symmetry of the pair of statistics (area, depth) for $\mathcal{K}$-Dyck paths. We provide an alternative description of the higher $q,t$-Catalan polynomials $C_{n}^{(k)}(q,t)$.

math.CO↗
arXiv2025

Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results

Cigler considered certain shifted Hankel determinants of convolution powers of Catalan numbers and conjectured identities for these determinants. Recently, Fulmek gave a bijective proof of Cigler's conjecture. Cigler then provided a computational proof. We extend Cigler's determinant identities to the convolution of general power series $F(x)$, where $F(x)$ satisfies a certain type of quadratic equation. As an application, we present the Hankel determinant identities of convolution powers of Motzkin numbers.

math.CO↗
arXiv2025

Hankel determinants for convolution powers of Motzkin numbers

We evaluate the Hankel determinants of the convolution powers of Motzkin numbers for $r\leq 27$ by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin's continued fraction method. We also conjecture some polynomial characterization of these determinants.

math.CO↗
arXiv2025

Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming

The Todd polynomials, denoted as $td_k(b_1,b_2,\ldots,b_m)$, are characterised by their generating functions: $$\sum_{k\ge 0} td_k s^k = \prod_{i=1}^m \frac{b_i s}{e^{b_i s}-1}.$$ These polynomials serve as fundamental components in the Todd class of toric varieties, a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalised Todd polynomials emerge naturally within the framework of MacMahon's partition analysis, particularly in the context of computing Ehrhart series. We introduce an efficient method for the evaluation of generalised Todd polynomials for numerical values of $b_i$. This is achieved through the development of expedited operations in the quotient ring $\mathbb{Z}_p[[s]]$ modulo $s^{d}$, where $p$ is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming when the dimension is fixed, exhibiting a notable enhancement in computational efficiency.

math.CO↗
arXiv2024

Inverting the General Order Sweep Map

Building upon the foundational work of Thomas and Williams on the modular sweep map, Garsia and Xin have developed a straightforward algorithm for the inversion of the sweep map on rational $(m,n)$-Dyck paths, where $(m,n)$ represents coprime pairs of integers. Our research reveals that their innovative approach readily generalizes to encompass a broader spectrum of Dyck paths. To this end, we introduce a family of Order sweep maps applicable to general Dyck paths, which are differentiated by their respective sweep orders at level $0$. We demonstrate that each of these Order sweep maps constitutes a bijective transformation. Our findings encapsulate the sweep maps for both general Dyck paths and their incomplete counterparts as specific instances within this more extensive framework.

math.CO↗
arXiv2024

A Combinatorial Decomposition of Knapsack Cones

In this paper, we focus on knapsack cones, a specific type of simplicial cones that arise naturally in the context of the knapsack problem $x_1 a_1 + \cdots + x_n a_n = a_0$. We present a novel combinatorial decomposition for these cones, named \texttt{DecDenu}, which aligns with Barvinok's unimodular cone decomposition within the broader framework of Algebraic Combinatorics. Computer experiments support us to conjecture that our \texttt{DecDenu} algorithm is polynomial when the number of variables $n$ is fixed. If true, \texttt{DecDenu} will provide the first alternative polynomial algorithm for Barvinok's unimodular cone decomposition, at least for denumerant cones. The \texttt{CTEuclid} algorithm is designed for MacMahon's partition analysis, and is notable for being the first algorithm to solve the counting problem for Magic squares of order 6. We have enhanced the \texttt{CTEuclid} algorithm by incorporating \texttt{DecDenu}, resulting in the \texttt{LLLCTEuclid} algorithm. This enhanced algorithm makes significant use of LLL's algorithm and stands out as an effective elimination-based approach.

math.CO↗
arXiv2024

Algebraic Volume for Polytope Arise from Ehrhart Theory

Volume computation for $d$-polytopes $\mathcal{P}$ is fundamental in mathematics. There are known volume computation algorithms, mostly based on triangulation or signed-decomposition of $\mathcal{P}$. We consider $ \mathrm{cone}(\mathcal{P})$ as a lift of $\mathcal{P}$ in view of Ehrhart theory. By using technique from algebraic combinatorics, we obtain a volume algorithm using only signed simplicial cone decompositions of $ \mathrm{cone}(¶)$. Each cone is associated with a simple algebraic volume formula. Summing them gives the volume of the polytope. Our volume formula applies to various kind of cases. In particular, we use it to explain the traditional triangulation method and Lawrence's signed decomposition method. Moreover, we give a completely new primal-dual method for volume computation. This solves the traditional problem in this area: All existing methods are hopelessly impractical for either the class of simple polytopes or the class of simplicial polytopes. Our method has a good performance in computer experiments.

math.CO↗
arXiv2021

Enumerating Magic Distinct Labellings of the Cube

We find by applying MacMahon's partition analysis that all magic labellings of the cube are of eight types, each generated by six basis elements. A combinatorial proof of this fact is given. The number of magic labellings of the cube is thus reobtained as a polynomial in the magic sum of degree $5$. Then we enumerate magic distinct labellings, the number of which turns out to be a quasi-polynomial of period 720720. We also find the group of symmetry can be used to significantly simplify the computation.

math.CO↗
arXiv2020

Dinv, Area, and Bounce for $\vec{k}$-Dyck paths

The well-known $q,t$-Catalan sequence has two combinatorial interpretations as weighted sums of ordinary Dyck paths: one is Haglund's area-bounce formula, and the other is Haiman's dinv-area formula. The zeta map was constructed to connect these two formulas: it is a bijection from ordinary Dyck paths to themselves, and it takes dinv to area, and area to bounce. Such a result was extended for $k$-Dyck paths by Loehr. The zeta map was extended by Armstrong-Loehr-Warrington for a very general class of paths. In this paper, We extend the dinv-area-bounce result for $\vec{k}$-Dyck paths by: i) giving a geometric construction for the bounce statistic of a $\vec{k}$-Dyck path, which includes the $k$-Dyck paths and ordinary Dyck paths as special cases; ii) giving a geometric interpretation of the dinv statistic of a $\vec{k}$-Dyck path. Our bounce construction is inspired by Loehr's construction and Xin-Zhang's linear algorithm for inverting the sweep map on $\vec{k}$-Dyck paths. Our dinv interpretation is inspired by Garsia-Xin's visual proof of dinv-to-area result on rational Dyck paths.

math.CO↗
arXiv2018

On the Sweep Map for $\vec{k}$-Dyck Paths

Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=kn\pm 1$. They introduced an intermediate family $\mathcal{T}_n^k$ of certain standard Young tableau. Then inverting the sweep map is done by a simple walking algorithm on a $T\in \mathcal{T}_n^k$. We find their idea naturally extends for $\mathbf{k}^\pm$-Dyck paths, and also for $\mathbf{k}$-Dyck paths (reducing to $k$-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in $\mathcal{T}_\mathbf{k}$ of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.

math.CO↗
arXiv2018

Modeling and Analysis of Cascading Failures in Interdependent Cyber-Physical Systems

Integrated cyber-physical systems (CPSs), such as the smart grid, are increasingly becoming the underpinning technology for major industries. A major concern regarding such systems are the seemingly unexpected large-scale failures, which are often attributed to a small initial shock getting escalated due to intricate dependencies within and across the individual counterparts of the system. In this paper, we develop a novel interdependent system model to capture this phenomenon, also known as cascading failures. Our framework consists of two networks that have inherently different characteristics governing their intra-dependency: i) a cyber-network where a node is functional as long as it belongs to the largest connected (i.e., giant) component; and ii) a physical network where nodes are given an initial flow and a capacity, and failure of a node results with redistribution of its flow to the remaining nodes, upon which further failures might take place due to overloading (i.e., the flow of a node exceeding its capacity). Furthermore, it is assumed that these two networks are inter-dependent. For simplicity, we consider a one-to-one interdependency model where every node in the cyber-network is dependent upon and supports a single node in the physical network, and vice versa. We provide a thorough analysis of the dynamics of cascading failures in this interdependent system initiated with a random attack. The system robustness is quantified as the surviving fraction of nodes at the end of cascading failures, and is derived in terms of all network parameters involved (e.g., degree distribution, load/capacity distribution, failure size, etc.). Analytic results are supported through an extensive numerical study. Among other things, these results demonstrate the ability of our model to capture the unexpected nature of large-scale failures and provide insights on improving system robustness.

physics.soc-ph↗
arXiv2018

Attack vulnerability of power systems under an equal load redistribution model

This paper studies the vulnerability of flow networks against adversarial attacks. In particular, consider a power system (or, any system carrying a physical flow) consisting of $N$ transmission lines with initial loads $L_1, \ldots , L_N$ and capacities $C_1, \ldots, C_N$, respectively; the capacity $C_i$ defines the maximum flow allowed on line $i$. Under an equal load redistribution model, where load of failed lines is redistributed equally among all remaining lines, we study the {\em optimization} problem of finding the best $k$ lines to attack so as to minimize the number of {\em alive} lines at the steady-state (i.e., when cascades stop). This is done to reveal the worst-case attack vulnerability of the system as well as to reveal its most vulnerable lines. We derive optimal attack strategies in several special cases of load-capacity distributions that are practically relevant. We then consider a modified optimization problem where the adversary is also constrained by the {\em total} load (in addition to the number) of the initial attack set, and prove that this problem is NP-Hard. Finally, we develop heuristic algorithms for selecting the attack set for both the original and modified problems. Through extensive simulations, we show that these heuristics outperform benchmark algorithms under a wide range of settings.

physics.soc-ph↗
arXiv2017

Cascading failures in interdependent systems under a flow redistribution model

Robustness and cascading failures in interdependent systems has been an active research field in the past decade. However, most existing works use percolation-based models where only the largest component of each network remains functional throughout the cascade. Although suitable for communication networks, this assumption fails to capture the dependencies in systems carrying a flow (e.g., power systems, road transportation networks), where cascading failures are often triggered by redistribution of flows leading to overloading of lines. Here, we consider a model consisting of systems $A$ and $B$ with initial line loads and capacities given by $\{L_{A,i},C_{A,i}\}_{i=1}^{n}$ and $\{L_{B,i},C_{B,i}\}_{i=1}^{n}$, respectively. When a line fails in system $A$, $a$-fraction of its load is redistributed to alive lines in $B$, while remaining $(1-a)$-fraction is redistributed equally among all functional lines in $A$; a line failure in $B$ is treated similarly with $b$ giving the fraction to be redistributed to $A$. We give a thorough analysis of cascading failures of this model initiated by a random attack targeting $p_1$-fraction of lines in $A$ and $p_2$-fraction in $B$. We show that (i) the model captures the real-world phenomenon of unexpected large scale cascades and exhibits interesting transition behavior: the final collapse is always first-order, but it can be preceded by a sequence of first and second-order transitions; (ii) network robustness tightly depends on the coupling coefficients $a$ and $b$, and robustness is maximized at non-trivial $a,b$ values in general; (iii) unlike existing models, interdependence has a multi-faceted impact on system robustness in that interdependency can lead to an improved robustness for each individual network.

physics.soc-ph↗
arXiv2016

Optimizing the robustness of electrical power systems against cascading failures

Electrical power systems are one of the most important infrastructures that support our society. However, their vulnerabilities have raised great concern recently due to several large-scale blackouts around the world. In this paper, we investigate the robustness of power systems against cascading failures initiated by a random attack. This is done under a simple yet useful model based on global and equal redistribution of load upon failures. We provide a complete understanding of system robustness by i) deriving an expression for the final system size as a function of the size of initial attacks; ii) deriving the critical attack size after which system breaks down completely; iii) showing that complete system breakdown takes place through a first-order (i.e., discontinuous) transition in terms of the attack size; and iv) establishing the optimal load-capacity distribution that maximizes robustness. In particular, we show that robustness is maximized when the difference between the capacity and initial load is the same for all lines; i.e., when all lines have the same redundant space regardless of their initial load. This is in contrast with the intuitive and commonly used setting where capacity of a line is a fixed factor of its initial load.

physics.soc-ph↗