Rational Combinatorics

Alexander Duality and Rational Associahedra

A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in turn a generalization of classical Catalan combinatorics). The latter paper gave two possible models for a rational analog of the associahedron which attach simplicial complexes to any pair of coprime positive integers a < b. These complexes coincide up to the Fuss-Catalan level of generality, but in general one may be a strict subcomplex of the other. Verif…

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart's method for proving that a couting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorithms for counting integer points in polyhedra and computing rational function representations, as well as visualizations …

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remov…

促进与行运动在有理Catalan组合学中

Promotion and rowmotion in rational Catalan combinatorics

本文研究了促进与行运动在有理Catalan组合学中的等价性，通过扩展双射与RSK类对应关系，揭示了 generalized Dyck paths 上的双射性质。

On the combinatorics of hypergeometric functions

We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.

Trees, graphs and aggregates: a categorical perspective on combinatorial surface topology, geometry, and algebra

Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical v…

Path combinatorics and light leaves for quiver Hecke algebras

We recast the classical notion of standard tableaux in an alcove-geometric setting and extend these classical ideas to all reduced paths in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias--Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the …

Highest weight sl_2-categorifications I: crystals

We define highest weight categorical actions of sl_2 on highest weight categories and show that basically all known examples of categorical sl_2-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories O of type $A$, categories O for cyclotomic Rational Cherednik algebras) are highest weight in our sense. Our main result is an explicit combinatorial description of (the labels of) the crystal on the set of simple objects. A new application of this is to determining the supports of simple modules over the cyclotomic Ra…

Combinatorics of the zeta map on rational Dyck paths

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of $P$ and zeta of $¶$ conjugate is enough to recover $P$. Our method begets an area-preserving involution $χ$ on the set of $(a,b)$-Dyck paths when $ζ$ is a bijection, as well as a new method for calculating $ζ^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck path…

根系偏序集中反链的单纯复形与Dyck路径的相关组合学

Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths

本文研究根系偏序集反链构成的单纯复形的可壳性，证明其等价于根系为A_n、B_n、D_3或G_2，并利用f-和h-三角形给出Dyck路径的统计量，有理情形留作开放问题。

On maximal rational poylhedral fans

In this paper we study the geometry and combinatorics of the possible rational polyhedral fans with a given set of rays. The main questions we consider are when such fans are projective, complete, or simplicial. To answer these questions we use techniques of invariant theory to characterize the quotients of maximal saturated open sets in torus representations.

Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by class…

Towards multiplicities for categories O of cyclotomic rational Cherednik algebras

Varagnolo and Vasserot conjectured an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a truncation of an affine parabolic category O of type A. In this paper we reduce their conjecture to some purely combinatorial conjecture. Our principal tool for the reduction is the theory of categorical sl_2-actions on highest weight categories. Then we prove our combinatorial conjecture in some cases.

Birational contractions of $\overline{\mathrm{M}}_{0,n}$ and combinatorics of extremal assignments

From Smyth's classification, modular compactifications of pointed smooth rational curves are indexed by combinatorial data, so-called extremal assignments. We explore their combinatorial structures and show that any extremal assignment is a finite union of atomic extremal assignments. We discuss a connection with the birational geometry of the moduli space of stable pointed curves. As applications, we study three special classes of extremal assignments: smooth, toric, and invariant with respect to the symmetric group action. We identify them with three combinatorial objects: simple intersectin…

The combinatorics of $\mathbb{C}^*$-fixed points in generalized Calogero-Moser spaces and Hilbert schemes

In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type $G(l,1,n)$, cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct $\mathbb{C}^*$-fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between $\mathbb{C}^*$-fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the $q$-ho…

Compositional Bernoulli numbers

We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics.

非齐次一阶迭代函数方程及其在组合学中的应用

Inhomogeneous order 1 iterative functional equations with applications to combinatorics

本文研究一阶迭代函数方程的解，证明在特定条件下解要么是有理函数或一阶线性微分方程的解，要么是微分超越的，并将该结果应用于三类组合生成函数的微分超越性判定。

Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

We consider the enumeration of walks on the non-negative lattice $\mathbb{N}^d$, with steps defined by a set $\mathcal{S} \subset \{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps $\mathcal{S}$ is symmetric over every axis; in this sett…

Rational Mixed Tate Motivic Graphs

In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in [12]). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles …

树管化与复苏型Dyson-Schwinger方程的组合学

Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations

该研究提出根树二元管化的组合学方法，为具有单一Mellin变换的Dyson-Schwinger方程提供级数解，并导出适合复苏分析的反常维数微分方程组。

Brézin--Gross--Witten 数的组合学与大亏格渐近

Combinatorics and large genus asymptotics of the Brézin--Gross--Witten numbers

本文研究 Brézin--Gross--Witten 数的组合与渐近性质，证明其归一化形式的一致大亏格首项渐近及多项式现象，并提出单调性与整性等新猜想，同时给出在 Painlevé II 层级和 BGW-kappa 数上的应用。

多面体的组合霍奇指标定理

Combinatorial Hodge Index Theorem for Polytopes

该研究将Maxim-Schuermann公式从顶点有理的多面体推广至任意凸多面体，依托BBFK的组合相交上同调框架，还探讨了多面体的霍奇指标定理。

A note on a local combinatorial formula for the Euler class of a PL spherical fiber bundle

We present a local combinatorial formula for the Euler class of a $n$-dimen\-si\-onal PL spherical fiber bundle as a rational number $e_{\it CH}$ associated to a chain of $n+1$ abstract subdivisions of abstract $n$-spherical PL cell complexes. The number $e_{\it CH}$ is a combinatorial (or matrix) Hodge theory twisting cochain in Guy Hirsch's homology model of the bundle associated with PL combinatorics of the bundle.

A local combinatorial formula for the Chern class of a triangulated $S^1$ bundle in terms of shellings

Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") of a triangulated $S^1$ bundle over a 2-simplex, which is a local combinatorial formula for the first Chern class. The curvature is expressed in terms of cyclic word in 3-character alphabet associated to the bundle. From the point of view of simplicial combinatorics the word is a …

Continued fractions and orderings on the Markov numbers

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbe…