Let $Φ$ be an irreducible crystallographic root system with Weyl group $W$ and coroot lattice $\check{Q}$, spanning a Euclidean space $V$. Let $m$ be a positive integer and $\aA^m_Φ$ be the arrangement of hyperplanes in $V$ of the form $(α, x) = k$ for $α\in Φ$ and $k = 0, 1,...,m$. It is known that the number $N^+ (Φ, m)$ of bounded dominant regions of $\aA^m_Φ$ is equal to the number of facets of the positive part $Δ^m_+ (Φ)$ of the generalized cluster complex associated to the pair $(Φ, m)$ by S. Fomin and N. Reading.
We define a statistic on the set of bounded dominant regions of $\aA^m_Φ$ and conjecture that the corresponding refinement of $N^+ (Φ, m)$ coincides with the $h$-vector of $Δ^m_+ (Φ)$. We compute these refined numbers for the classical root systems as well as for all root systems when $m=1$ and verify the conjecture when $Φ$ has type $A$, $B$ or $C$ and when $m=1$. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of $Φ$, orbits of the action of $W$ on the quotient $\check{Q} / (mh-1) \check{Q}$ and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of $\aA^m_Φ$. We also provide a dual interpretation in terms of order filters in the root poset of $Φ$ in the special case $m=1$.
We characterize in simple terms the Pak-Stanley labels $λ(R)$ of the regions $R$ of the $m$-Catalan arrangement. We also propose a simple algorithm that returns $R$ from $λ(R)$. Finally, we characterize in close terms the labels of the relatively bounded regions.
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.
The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended Catalan arrangements, using species. In this article, we introduce the species of flats of the extended Catalan and Shi arrangements and they are given by iterated substitution of species of sets and lists. Moreover, we enumerate the flats of these arrangements in terms of infinite matrices.
The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give addition…
Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric function…
Let $W$ be a finite Weyl group and $\A$ be the corresponding Weyl arrangement. A deformation of $\A$ is an affine arrangement which is obtained by adding to each hyperplane $H\in\A$ several parallel translations of $H$ by the positive root (and its integer multiples) perpendicular to $H$. We say that a deformation is $W$-equivariant if the number of parallel hyperplanes of each hyperplane $H\in \A$ depends only on the $W$-orbit of $H$. We prove that the conings of the $W$-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers.…
The paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of the paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers…
We give a complete formula for the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the hyperplanes $x_i+x_j=1$, $x_k=0$, $x_l=1$, $ 1\leq i, j, k, l\leq n$. The formula is obtained by associating hyperplane arrangements with graphs, and then enumerating central graphs via generating functions for the number of bipartite graphs of given order, size and number of connected components.
We present determinantal representations of the Catalan numbers, k-Fuss-Catalan numbers, and its generalized number. The entries of the normalized Hessenberg matrices are the binomial coefficients that related with the enumeration of lattice paths.
A non-crossing pairing on a bitstring matches 1s and 0s in a manner such that the pairing diagram is nonintersecting. By considering such pairings on arbitrary bitstrings $1^{n_1} 0^{m_1} ... 1^{n_r} 0^{m_r}$, we generalize classical problems from the theory of Catalan structures. In particular, it is very difficult to find useful explicit formulas for the enumeration function $ϕ(n_1, m_1, ..., n_r, m_r)$, which counts the number of pairings as a function of the underlying bitstring. We determine explicit formulas for $ϕ$, and also prove general upper bounds in terms of Fuss-Catalan numbers by…
In this paper, we give a bijection between rooted labeled ordered forests with a selected subset of their leaves and the regions of the type $C$ Catalan arrangement in $\R^n$. We thus obtain a bijective proof of the well-known enumeration formula of these regions ${2^n}n! \binom{2n}{n}$.
This article introduces and investigates a refinement of alternating sign trapezoids by means of Catalan objects and Motzkin paths. Alternating sign trapezoids are a generalisation of alternating sign triangles, which were recently introduced by Ayyer, Behrend and Fischer. We show that the number of alternating sign trapezoids associated with a Catalan object (resp. Motzkin path) is a polynomial function in the length of the shorter base of the trapezoid. We also study the rational roots of the polynomials and formulate several conjectures and derive some partial results. Lastly, we deduce a c…
We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules. We focus on the standard puzzles for which the enumeration problems can be solved by explicit formulas or by classical numbers, such as binomial coefficients, Fibonacci numbers, tangent numbers, Catalan numbers, $\ldots$
We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.
The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $Φ$ is a well-studied object enumerated by the Fuß-Catalan number $Cat^{(k)}(Φ)$. It is natural to refine this enumeration by considering floors and ceilings of dominant regions. A conjecture of Armstrong states that counting dominant regions by their number of floors of a certain height gives the same distribution as counting dominant regions by their number of ceilings of the same height. We prove this conjecture using a bijection that provides even more refined enumerative information.
In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.
We introduce a combinatorial enumeration problem that is solved using generalized Catalan numbers. We also study generalizations of the Cycle Lemma beyond the computation of the generalized Catalan numbers.
We work with lattice walks in $\mathbb{Z}^{r+1}$ using step set $\{\pm 1\}^{r+1}$ that finish with $x_{r+1} = 0$. We further impose conditions of avoiding backtracking (i.e. $[v,-v]$) and avoiding consecutive steps (i.e. $[v,v]$) each possibly combined with restricting to the half-space $x_{r+1} \geq 0$. We find in all cases the generating functions for such walks are algebraic and give explicit formulas for them. We also find polynomial recurrences for their coefficients. From the generating functions we find the asymptotic enumeration of each family of walks considered. The enumeration in sp…
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal structure of a graph, and contains even other specializations from other sciences like the Jones polynomial in Knot theory, the partition function of the Pott model in statistical physics, and the reliability polynomial in network theory. In this article, we study the Tutte polynomial associated to more general objects which are the arrangements of hyperplanes.…
We explain how the moments of the (weight function of the) Askey Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors. This gives us a direct combinatorial formula for these moments. Then we use techniques developed by Ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux. Finally we study the enumeration of staircase tableaux at various specializations of the parameterizations; for example, we obtain the Catalan numbers, Fibonacci numbers, Eulerian numbers, the num…
Standard set-valued Young tableaux are a generalization of standard Young tableaux in which cells may contain more than one integer, with the added conditions that every integer at position $(i,j)$ must be smaller than every integer at positions $(i,j+1)$ and $(i+1,j)$. This paper explores the combinatorics of standard set-valued Young tableaux with two-rows, and how those tableaux may be used to provide new combinatorial interpretations of generalized Catalan numbers. New combinatorial interpretations are provided for the two-parameter Fuss-Catalan numbers (Raney numbers), the rational Catala…
The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families $F_1 \subset F_2 \subset F_3 \subset F_4$ of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family $F_i…
In this paper we present a bijection $ω_n$ between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $Δ^m(A_n)$ and the set of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, $ω_n$ bijects the facets containing the negative simple root $-α$ to dominant regions having the hyperplane $\{v\in V\mid<v,α>=m\}$ as separating wall. As a result, $ω_n$ restricts to a bijection between the set of facets of the positive part of $Δ^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}…
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated discriminantal arrangement on the other hand. The type of hyperplane arrangements considered and the isomorphism classes have been defined precisely. As a consequence we enumerate such isomorphism classes by computing the characteristic polynomial of the discriminantal arrangement. With a certain restriction, the enumerated value is shown to be independent of the discri…
For a crystallographic root system, dominant regions in the Catalan hyperplane arrangement are in bijection with antichains in a partial order on the positive roots. For a noncrystallographic root system, the analogous arrangement and regions have importance in the representation theory of an associated graded Hecke algebra. Since there is also an analogous root order, it is natural to hope that a similar bijection can be used to understand these regions. We show that such a bijection does hold for type $H_3$ and for type $I_2(m)$, including arbitrary ratio of root lengths when $m$ is even, bu…
A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws lead to the "weak chromatic group" of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral…
A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that …
We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of $sl(n+1,\Bbb C)$. We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q,t)-analogue of the Catalan number $C_n$. These (q,t)-Catalan numbers count on the one hand ad-nilpotent ideals with respect to dimension and class of nilpote…
We find a new approach to computing the remainder of a polynomial modulo $x^n-1$; such a computation is called modular enumeration. Given a polynomial with coefficients from a commutative $\mathbb{Q}$-algebra, our first main result constructs the remainder simply from the coefficients of residues of the polynomial modulo $Φ_d(x)$ for each $d\mid n$. Since such residues can often be found to have nice values, this simplifies a number of modular enumeration problems; indeed in some cases, such residues are already known while the related modular enumeration problem has remained unsolved. We list…
We consider the problem of counting the set of $\mathscr{D}_{a,b}$ of Dyck paths inscribed in a rectangle of size $a\times b$. They are a natural generalization of the classical Dyck words enumerated by the Catalan numbers. By using Ferrers diagrams associated to Dyck paths, we derive formulas for the enumeration of $\mathscr{D}_{a,b}$ with $a$ and $b$ non relatively prime, in terms of Catalan numbers.
We establish counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x\_i-x\_j=s$ for some integer $s$. Classical examples include the braid, Catalan, Shi, semiorder and Linial arrangements, as well as graphical arrangements. We express the number of regions of any such arrangement as a signed count of decorated plane trees. The characteristic and coboundary polynomials of these arrangements also have simple expressions in terms of these trees. We then focus on certain "well-b…
In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection be…
Dyck paths (also balanced brackets and Dyck words) are among the most heavily studied Catalan families. This paper is a continuation of [2, 3]. In the paper we enumerate the terms of the OEIS A036991, Dyck numbers, and construct a concomitant bijection with symmetric Dyck paths. In the case of binary coding of Dyck paths we work with compact natural numbers after removing leading zeros. Analysis of binary suffixes, allowed us to obtain a bijection between arbitrary A036991 terms and symmetric A036991 terms which encode symmetric Dyck paths. The bijection generates a forest of unary non-interse…
A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the $n=3$ case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane arrangements and Ehrhart's theory of lattice-point enumeration.
The Raney numbers $R_{p,r}(k)$ are a two-parameter generalization of the Catalan numbers. In this paper, we obtain a recurrence relation for the Raney numbers which is a generalization of the recurrence relation for the Catalan numbers. Using this recurrence relation, we confirm a conjecture posed by Amdeberhan concerning the enumeration of $(s,s+1)$-core partitions $λ$ with parts that are multiples of $p$. We then give a new combinatorial interpretation for the Raney numbers $R_{p+1,r+1}(k)$ with $0\leq r<p$ in terms of $(kp+r,kp+r+1)$-core partitions $λ$ with parts that are multiples of $p$.
In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem. Recently, the multiple version of the addition theorem is proved, called the multiple addition theorem (MAT) to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT from the viewpoint of our new proof. Moreover, we introduce their restriction version, a multiple restriction theorem (MRT). Applications of them including the combinatorial freeness of the extended C…
A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\bf C}_g(z)=\sum_{n\geq 0} {\bf c}_g(n)z^n$ for the number ${\bf c}_g(n)$ of linear chord diagrams of fixed genus $g\geq 1$ with a given number $n\geq 0$ of chords and find the remarkably simple formula ${\bf C}_g(z)=z^{2g}R_g(z) (1-4z)^{{1\over 2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$ with integral coefficients satisfying $R_g({1\over 4})\neq 0$ and $R_g(0) = {\bf c}_g(2g)\neq 0.$ In particular, ${\bf C}_g(z)$ is algebraic over $\m…
The Catalan numbers $C_k$ were first studied by Euler, in the context of enumerating triangulations of polygons $P_{k+2}$. Among the many generalizations of this sequence, the Fuss-Catalan numbers $C^{(d)}_k$ count enumerations of dissections of polygons $P_{k(d-1)+2}$ into $(d+1)$-gons. In this paper, we provide a formula enumerating polygonal dissections of $(n+2)$-gons, classified by partitions $λ$ of $[n]$. We connect these counts $a_λ$ to reverse series arising from iterated polynomials. Generalizing this further, we show that the coefficients of the reverse series of polynomials $x=z-\su…
We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface $Σ$ and introduce the number $C_Σ(n)$ of non-crossing partitions of a set of $n$ points laying on the boundary of $Σ$. Our proofs use bijective techniques arising from map enumeration, joint with the symbolic method and singularity analysis on generating functions. An outcome of our results is that the exponential growth of $C_Σ(n)$ is the same as the one of the $n$-th Catalan number, i.e., does not change when we move from the case where $Σ$ is a disk to general surf…
In 2003, Haglund's {\sf bounce} statistic gave the first combinatorial interpretation of the $q,t$-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type $A$. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement - which we call the Ish arrangement. We prove that our statistics are equivalent to the {\sf area'} and {\sf bounce} statistics of Haglund and Loehr. In this setting, we observe that…
In 1995, the first author introduced a multivariate generating function {$G$} that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that $G$ is symmetric, he conjectured that $G$ is Schur positive. We prove this conjecture by expanding $G$ positively in terms of ribbon Schur functions. We obtain this expansion using a weight-preserving bijection whose inverse is inspired by the Push-Glide algorithm of Préville-Ratelle and Viennot. In fact, this weight-preserving bijection allows us to establish a stronger version of the first author's conjecture s…
A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytop…
An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the characteristic polynomial of the Shi threshold and the Catalan threshold arrangements was an open problem in Stanley's list of problems in [1]. Seunghyun Seo solved both the problems by clever arguments using the finite field method in [3,4]. However, in his paper, he left open the problem of computing the characteristic polynomial of a broader class of thr…
