Christos A. Athanasiadis
We prove a conjecture of F. Chapoton relating certain enumerative invariants of (a) the cluster complex associated by S. Fomin and A. Zelevinsky to a finite root system and (b) the lattice of noncrossing partitions associated to the corresponding finite real reflection group.
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Some enumerative aspects of the fans, called generalized associahedra, introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras are considered, in relation with a bicomplex and its two spectral sequences. A precise enumerative relation with the lattices of generalized noncrossing partitions is conjectured and some evidence is given.
We continue the study of the nonkissing complex that was introduced by Petersen, Pylyavskyy, and Speyer and was studied lattice-theoretically by the second author. We introduce a theory of Grid-Catalan combinatorics, given the initial data of a nonkissing complex, and show how this theory parallels the well-known Coxeter-Catalan combinatorics. In particular, we present analogues of Chapoton's F-triangle, H-triangle, and M-triangle and give combinatorial, lattice-theoretic, and geometric interpretations of the objects defining these. In our Grid-Catalan setting, we prove that Chapoton's F-trian…
Root systems and generalized associahedra
本文综述了根系统、广义阿斯托哈德拉及簇的组合学，涵盖经典根系统、反射群及簇代数的 combinatorial 方法。
Enumerative aspects of Caylerian polynomials
本研究聚焦Caylerian多项式的计数问题，类比欧拉多项式的下降位分布定义，结合组合物种与符号反转对合方法，推导出该多项式及相关精细化多项式的计数公式与生成函数。
This paper introduces an inductively defined tree notation for all the faces of polytopes arising from a simplex by truncations. This notation allows us to view inclusion of faces as the process of contracting tree edges. Our notation instantiates to the well-known notations for the faces of associahedra and permutohedra. Various authors have independently introduced combinatorial tools for describing such polytopes. We build on the particular approach developed by Dosen and Petric, who used the formalism of hypergraphs to describe the interval of polytopes from the simplex to the permutohedro…
Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by systems of polynomial equations satisfying special properties. If the equations are infeasible, Hilbert's Nullstellensatz gives a certificate of this fact. These certificates have been studied and exhibit combinatorial meaning. In this paper, we generalize some known results and show that the Nullstellensatz certificate can be viewed as enumerating combinatorial …
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called constellations provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumer…
We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.
Lattice path matroids: enumerative aspects and Tutte polynomials
本文定义格路拟阵，研究其枚举性质及 Tutte 多项式、特征多项式与 beta 不变量，并证明 Tutte 多项式可多项式时间计算。
We focus on combinatorial aspects of the Hilbert series of the cohomology ring of the moduli space of stable pointed curves of genus zero. We show its graded Hilbert series satisfies an integral operator identity. This is used to give asymptotic behavior, and in some cases, exact values, of the coefficients themselves. We then study the total dimension, that is, the sum of the coefficients of the Hilbert series. Its asymptotic behavior involves the Lambert W function, which has applications to classical tree enumeration, signal processing and fluid mechanics.
This manuscript recounts some of the author's contributions to algebraic and enumerative combinatorics. We have focused on two types of generalizations of bipartite maps, which are bipartite graphs embedded on surfaces. Maps are known to appear in many areas of theoretical physics and discrete mathematics, but one key interest for fundamental computer science is how multi-facetted they are in the sense that multiple encodings exist which are not interchangeable, like Tutte's/loop equations, the topological recursion, the KP hierarchy and numerous bijections which made the field so rich. One ge…
For the word $ω= \underbrace{11\ldots 1}_{x_1}\underbrace{22\ldots2}_{x_2}\ldots\underbrace{nn\ldots n}_{x_n},$ denote by $\mathsf{A}(x_1, x_2, \ldots, x_n)$ the number of its anagrams without fixed letters. While the function $\mathsf{A}()$ bears significant importance to economic theory \cite{MCKELVEY1997411}, it is not known whether it can be computed in polynomial time. The desire to answer efficiently certain queries related to this function motivates our study of its combinatorial properties. Our first main result shows that $\mathsf{A}(x_1, x_2, \ldots, x_n)\pmod{p}$ can be efficiently …
Algorithmic and enumerative aspects of the Moser-Tardos distribution
本文研究Moser-Tardos算法中间及输出分布的随机性，证明其可加速坏事件查找、改进算法复杂度，并利用最小熵证明枚举问题解的数量更多，同时提出在LLL条件违反时仍有效的变体。
Generalized cluster complexes and Coxeter combinatorics
本文介绍了一类与有限根系和非负整数参数m相关的简单复形，通过计算其面数和h向量，揭示了与Coxeter群、根系和超平面排列的联系，并提出基于递推的算法计算Coxeter不变量。
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