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Jianping Pan

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math.GM math.CO Networking and Internet Architecture Machine Learning Computational Geometry math.QA

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preprint2026arXiv

Hook-valued tableau uncrowding and tableau switching

Refined canonical stable Grothendieck polynomials were introduced by Hwang, Jang, Kim, Song, and Song. There exist two combinatorial models for these polynomials: one using hook-valued tableaux and the other using pairs of a semistandard Young tableau and (what we call) an exquisite tableau. An uncrowding algorithm on hook-valued tableaux was introduced by Pan, Pappe, Poh, and Schilling. In this paper, we discover a novel connection between the two models via the uncrowding and Goulden--Greene's jeu de taquin algorithms, using a classical result of Benkart, Sottile, and Stroomer on tableau switching. This connection reveals a symmetry of the uncrowding algorithm defined on hook-valued tableaux. As a corollary, we obtain another combinatorial model for the refined canonical stable Grothendieck polynomials in terms of biflagged tableaux, which naturally appear in the characterization of the image of the uncrowding map.

preprint2026arXiv

Worst-Case Regret Bounds for Combinatorial Thompson Sampling in Sleeping Semi-Bandits

We revisit combinatorial Thompson sampling (CTS) for semi-bandits with sleeping arms, where arm availability varies over time and actions must satisfy combinatorial constraints, as in wireless mesh routing with fluctuating link availability. Despite its practical relevance, CTS has been hindered by several long-standing problems: (i) the absence of worst-case regret guarantees in the semi-bandit setting even without sleeping arms, (ii) the lack of theory under adversarially varying availability, and (iii) the consistently weak empirical performance of CTS with Gaussian priors (CTS-G). This paper resolves these long-standing issues by providing the first worst-case regret analysis of CTS-G, proving an upper bound of $\tilde{O}(m\sqrt{NT})$ and a matching lower bound of $\tildeΩ(m\sqrt{NT})$. To bridge the gap between theory and practice, we further propose CL-SG, a simple CTS-G variant that samples a single shared Gaussian seed each round to coordinate exploration across arms. We show that CL-SG achieves an improved regret bound of $\tilde{O}(\sqrt{mNT})$, together with a matching lower bound $Ω(\sqrt{mNT})$. Experiments on real-world datasets demonstrate that CL-SG consistently outperforms strong baselines including CTS-G and CTS-B, and we open-source our implementation for reproducibility.

preprint2022arXiv

A bijection between $K$-Kohnert diagrams and reverse set-valued tableaux

Lascoux polynomials are $K$-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux ($\mathsf{RSVT}$) rule for Lascoux polynomials and reverse semistandard Young tableaux ($\mathsf{RSSYT}$) rule for key polynomials. Furthermore, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with $\mathsf{RSSYT}$. Ross and Yong introduced $K$-Kohnert diagrams, which are analogues of Kohnert diagrams. They conjectured a $K$-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between $\mathsf{RSVT}$ and $K$-Kohnert diagrams.

preprint2021arXiv

Random Distances Associated with Hexagons

In this report, the explicit probability density functions of the random Euclidean distances associated with regular hexagons are given, when the two endpoints of a link are randomly distributed in the same hexagon, and two adjacent hexagons sharing a side, respectively. Simulation results show the accuracy of the obtained closed-form distance distribution functions, which are important in a wide variety of applied sciences and engineering fields. In particular, hexagons are often used in wireless communication networks such as the cellular systems. The correctness of these distance distribution functions is validated by a recursion and a probabilistic sum. The first two statistical moments of the random distances, and the polynomial fits of the density functions are also given in this report for practical uses.

preprint2020arXiv

A crystal on decreasing factorizations in the $0$-Hecke monoid

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.

preprint2020arXiv

Thompson Sampling for Combinatorial Semi-bandits with Sleeping Arms and Long-Term Fairness Constraints

We study the combinatorial sleeping multi-armed semi-bandit problem with long-term fairness constraints~(CSMAB-F). To address the problem, we adopt Thompson Sampling~(TS) to maximize the total rewards and use virtual queue techniques to handle the fairness constraints, and design an algorithm called \emph{TS with beta priors and Bernoulli likelihoods for CSMAB-F~(TSCSF-B)}. Further, we prove TSCSF-B can satisfy the fairness constraints, and the time-averaged regret is upper bounded by $\frac{N}{2η} + O\left(\frac{\sqrt{mNT\ln T}}{T}\right)$, where $N$ is the total number of arms, $m$ is the maximum number of arms that can be pulled simultaneously in each round~(the cardinality constraint) and $η$ is the parameter trading off fairness for rewards. By relaxing the fairness constraints (i.e., let $η\rightarrow \infty$), the bound boils down to the first problem-independent bound of TS algorithms for combinatorial sleeping multi-armed semi-bandit problems. Finally, we perform numerical experiments and use a high-rating movie recommendation application to show the effectiveness and efficiency of the proposed algorithm.

preprint2016arXiv

Random Distances Associated with Arbitrary Polygons: An Algorithmic Approach between Two Random Points

This report presents a new, algorithmic approach to the distributions of the distance between two points distributed uniformly at random in various polygons, based on the extended Kinematic Measure (KM) from integral geometry. We first obtain such random Point Distance Distributions (PDDs) associated with arbitrary triangles (i.e., triangle-PDDs), including the PDD within a triangle, and that between two triangles sharing either a common side or a common vertex. For each case, we provide an algorithmic procedure showing the mathematical derivation process, based on which either the closed-form expressions or the algorithmic results can be obtained. The obtained triangle-PDDs can be utilized for modeling and analyzing the wireless communication networks associated with triangle geometries, such as sensor networks with triangle-shaped clusters and triangle-shaped cellular systems with highly directional antennas. Furthermore, based on the obtained triangle-PDDs, we then show how to obtain the PDDs associated with arbitrary polygons through the decomposition and recursion approach, since any polygons can be triangulated, and any geometry shapes can be approximated by polygons with a needed precision. Finally, we give the PDDs associated with ring geometries. The results shown in this report can enrich and expand the theory and application of the probabilistic distance models for the analysis of wireless communication networks.

preprint2015arXiv

Recursion-based Analysis for Information Propagation in Vehicular Ad Hoc Networks

Effective inter-vehicle communication is fundamental to a decentralized traffic information system based on Vehicular Ad Hoc Networks (VANETs). To reflect the uncertainty of the information propagation, most of the existing work was conducted by assuming the inter-vehicle distance follows some specific probability models, e.g., the lognormal or exponential distribution, while reducing the analysis complexity. Aimed at providing more generic results, a recursive modeling framework is proposed for VANETs in this paper when the vehicle spacing can be captured by a general i.i.d. distribution. With the framework, the analytical expressions for a series of commonly discussed metrics are derived respectively, including the mean, variance, probability distribution of the propagation distance, and expectation for the number of vehicles included in a propagation process, when the transmission failures are mainly caused by MAC contentions. Moreover, a discussion is also made for demonstrating the efficiency of the recursive analysis method when the impact of channel fading is also considered. All the analytical results are verified by extensive simulations. We believe that this work is able to potentially reveal a more insightful understanding of information propagation in VANETs by allowing to evaluate the effect of any vehicle headway distributions.

preprint2014arXiv

A Geometrical-Based Throughput Bound Analysis for Device-to-Device Communications in Cellular Networks

Device-to-device (D2D) communications in cellular networks are promising technologies for improving network throughput, spectrum efficiency, and transmission delay. In this paper, we first introduce the concept of guard distance to explore a proper system model for enabling multiple concurrent D2D pairs in the same cell. Considering the Signal to Interference Ratio (SIR) requirements for both macro-cell and D2D communications, a geometrical method is proposed to obtain the guard distances from a D2D user equipment (DUE) to the base station (BS), to the transmitting cellular user equipment (CUE), and to other communicating D2D pairs, respectively, when the uplink resource is reused. By utilizing the guard distances, we then derive the bounds of the maximum throughput improvement provided by D2D communications in a cell. Extensive simulations are conducted to demonstrate the impact of different parameters on the optimal maximum throughput. We believe that the obtained results can provide useful guidelines for the deployment of future cellular networks with underlaying D2D communications.

preprint2013arXiv

Random Distances Associated with Trapezoids

The distributions of the random distances associated with hexagons, rhombuses and triangles have been derived and verified in the existing work. All of these geometric shapes are related to each other and have various applications in wireless communications, transportation, etc. Hexagons are widely used to model the cells in cellular networks, while trapezoids can be utilized to model the edge users in a cellular network with a hexagonal tessellation. In this report, the distributions of the random distances associated with unit trapezoids are derived, when two random points are within a unit trapezoid or in two neighbor unit trapezoids. The mathematical expressions are verified through simulation. Further, we present the polynomial fit for the PDFs, which can be used to simplify the computation.

preprint2012arXiv

Random Distances Associated With Equilateral Triangles

In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent triangles sharing a side, 3) two parallel triangles sharing a vertex, and 4) two diagonal triangles sharing a vertex, respectively. The density function for 1) is based on the most recent work by Basel [1]. 2)-4) are based on 1) and our previous work in [2], [3]. Simulation results show the accuracy of the obtained closed-form distance distribution functions, which are important in the theory of geometrical probability. The first two statistical moments of the random distances and the polynomial fits of the density functions are also given in this report for practical uses.

preprint2011arXiv

Random Distances Associated with Rhombuses

Parallelograms are one of the basic building blocks in two-dimensional tiling. They have important applications in a wide variety of science and engineering fields, such as wireless communication networks, urban transportation, operations research, etc. Different from rectangles and squares, the coordinates of a random point in parallelograms are no longer independent. As a case study of parallelograms, the explicit probability density functions of the random Euclidean distances associated with rhombuses are given in this report, when both endpoints are randomly distributed in 1) the same rhombus, 2) two parallel rhombuses sharing a side, and 3) two rhombuses having a common diagonal, respectively. The accuracy of the distance distribution functions is verified by simulation, and the correctness is validated by a recursion and a probabilistic sum. The first two statistical moments of the random distances, and the polynomial fit of the density functions are also given in this report for practical uses.