François Bergeron
We formulate many open questions regarding the Schur positivity of the effect of interesting operators on symmetric functions, and give supporting evidence for why one should expect such behavior.
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We solve two open problems in Coxeter-Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type-uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter-Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.
Rectangular Symmetry for Catalan Functions
该研究针对卡塔兰函数的舒尔系数，通过证明表对合的性质，给出了其矩形对称性的双射证明。
We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps and explicit computations of the asymptotic expansion of the partition function of random matrices in the unitary ensemble case. We give combinatoric proofs of the formulas for the number of genus 0 and genus 1 maps.
While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to a refined understanding of the Kronecker squares. Furthermore, connections to 2-modular decomposition numbers, Catalan combinatorics, and to the Saxl conjecture are discussed which further motivate the study of these splittings.
Our main aim with these notes is to introduce the combinatorial and symmetric function tools that relate to the description of the Poincare polynomial of the triply graded Khovanov-Rozansky homology of torus links, a.k.a. the (reduced) superpolynomial of these links.
We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.
The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition $s$ which is motivated by the combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$ we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair $(a,b)$ of relatively prime numbers we can asso…
Combinatorics of three-Catalan numbers and some positivities
本文提出三Catalan三角，利用三Catalan数的递推与组合解释证明其对数凸性，并证明三角各行具有对数凹性。
Symmetric functions provide one of the most efficient tools for combinatorial enumeration, in the context of objects that may be acted upon by permutations. Only assuming a basic knowledge of linear algebra, we introduce and describe the classical toolbox of symmetric functions, and give many examples of their uses. In particular, we present how classical formulas of enumerative combinatorics afford a natural generalization in terms of symmetric functions. We then rapidly present Macdonald symmetric functions and associated operators, to go on with the description of their relations to rectang…
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
Major Index on Catalan Combinatorics
本文研究了 $r$-Dyck 路径等 Catalan 组合对象上的逆序数和 major index 统计量，并证明了它们与 Dyck 路径上的对应统计量具有等分布性。
In this paper we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results…
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuss-Catalan combinatorics. Having developed the combinatorial theory, we…
In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations.
In this paper we determine the parity of some sequences which are related to Catalan numbers. Also we introduce a combinatorical object called, \Catalan tree", and discuss its properties.
Set-Valued Catalan Combinatorics
本文研究两行集值标准杨表的计数，证明其为Catalan等数的新组合模型，还推广集值余主指标得到新的q-模拟。
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…
This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories…
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This paper proves the equivalence of the two definitions for all $m\geq 1$ and all $n\leq 4$. We also give a bijective proof of the joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $m\geq 1$ and all $n\leq 4$. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes…
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。
We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our b…
We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of (m,n)-parking functions, and associated (q,t)-analogs.
We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible applications to Interaction-Round-a-Face and hard-particle statistical models defined on random lattices.
Coxeter-biCatalan combinatorics
本文探讨Coxeter-Catalan组合学中的计数问题，提出双对象计数方法，证明所有问题有相同解，并定义W-biCatalan数。
A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over Z^M, and on the other hand, they count integer …
The Catalan number sequence is one of the most famous number sequences in combinatorics and is well studied in the literature. In this paper we further investigate its fundamental properties related to the moment problem and prove for the first time that it is an infinitely divisible Stieltjes moment sequence in the sense of S.-G. Tyan. Besides, any positive real power of the sequence is still a Stieltjes determinate sequence. Some more cases including (a) the central binomial coefficient sequence (related to the Catalan sequence), (b) a double factorial number sequence and (c) the generalized…
Logarithms of Catalan generating functions: A combinatorial approach
本文研究了第k个广义Catalan数生成函数G_k取对数操作的组合学，通过格路径和树路径提供组合解释，并利用显式双射以纯组合枚举手段恢复了log G_k系数的已知表达式，且该非代数证明可推广至更高次幂。
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.
Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube
本文研究了超立方体划分的枚举问题，通过正交划分和满$p$-ary树的双射，推导了卡特兰数的广义渐进行为公式。
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…
This is the first in a series of papers in which we describe explicit structural properties of spaces of diagonal rectangular harmonic polynomials in $k$ sets of $n$ variables, both as $GL_k$-modules and $S_n$-modules, as well as some of there relations to areas such as Algebraic Combinatorics, Representation Theory, Algebraic Geometry, Knot Theory, and Theoretical Physics. Our global aim is to develop a unifying point of view for several areas of research of the last two decades having to do with Macdonald Polynomials Operator Theory, Diagonal Coinvariant Spaces, Rectangular-Catalan Combinato…
Residue Formulae for the Trace on Affine Hecke Algebras
受Catalan组合学启发，利用Szenes-Vergne理论从Opdam生成函数导出仿射Hecke代数标准迹的留数公式，进而得到大腔中平移元素迹的乘积公式。
A recent approach to the Beck-Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoss (SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main tool is a conditional second moment method combined with Stein's method of exch…
There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have previously appeared in the literature. Concretely, for every complex polynomial $p$ with $d$ distinct roots and degree at least 2, we produce a canonical compact planar 2-complex that is a compact metric version of a tiled phase diagram. It has a locally CAT(0) metric that is locally Euclidean away from a finite set of interior points indexed by the critical points o…
In How Many Ways can a Rectangle be Rectangled?
本文通过两种新方法推广了Klarner-Magliveras和Smith-Verrill关于矩形网格矩形剖分计数的工作，实现了加权计数并统计了瓷砖数量和网格边数的分布。
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