Catalan Numbers Research Articles

A New Approach to k-Oresme and k-Oresme-Lucas Sequences

In this study, the k-Oresme and k-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the k-Oresme and k-Oresme-Lucas sequences are presented. In addition, we give these sequences the Binet formulas, generating functions, Cassini identity, Catalan identity etc. Moreover, the k-Oresme and k-Oresme-Lucas sequences are associated with Fibonacci, Pell numbers and Lucas, and Pell- Lucas numbers, respectively. Finally, the Catalan transforms of these sequences are given and Hankel transforms are applied to these Catalan sequences and associated with the terms of the sequence.

Quantum lozenge tiling and entanglement phase transition

While volume violation of area law has been exhibited in several quantum spin chains, the construction of a corresponding ground state in higher dimensions, entangled in more than one direction, has been an open problem. Here we construct a 2D frustration-free Hamiltonian with maximal violation of the area law. We do so by building a quantum model of random surfaces with color degree of freedom that can be viewed as a collection of colored Dyck paths. The Hamiltonian may be viewed as a 2D generalization of the Fredkin spin chain. It relates all the colored random surface configurations subject to a Dirichlet boundary condition and hard wall constraint from below to one another, and the ground state is therefore a superposition of all such classical states and non-degenerate. Its entanglement entropy between subsystems undergoes a quantum phase transition as the deformation parameter is tuned. The area- and volume-law phases are similar to the one-dimensional model, while the critical point scales with the linear size of the system L as Llog⁡L. Further it is conjectured that similar models with entanglement phase transitions can be built in higher dimensions with even softer area law violations at the critical point.

Schröder–Catalan Matrix and Compactness of Matrix Operators on Its Associated Sequence Spaces

In this article, the regular Schröder–Catalan matrix is constructed and acquired by benefiting Schröder and Catalan numbers. After that, two sequence spaces are introduced, described as the domain of Schröder–Catalan matrix. Additionally, some algebraic and topological properties of the spaces in question, such as completeness, inclusion relations, basis and duals, are examined. In the last two sections, the necessary and sufficient conditions of some matrix classes and compact operators related aforementioned spaces are presented.

The excedance quotient of the Bruhat order, quasisymmetric varieties, and Temperley–Lieb algebras

AbstractLet be the ring of polynomials in variables and consider the ideal generated by quasisymmetric polynomials without constant term. It was shown by J. C. Aval, F. Bergeron, and N. Bergeron that the th Catalan number. In the present work, we explain this phenomenon by defining a set of permutations with the following properties: first, is a basis of the Temperley–Lieb algebra , and second, when considering as a collection of points in , the top‐degree homogeneous component of the vanishing ideal is . Our construction has a few byproducts that are independently noteworthy. We define an equivalence relation on the symmetric group using weak excedances and show that its equivalence classes are naturally indexed by noncrossing partitions. Each equivalence class is an interval in the Bruhat order between an element of and a 321‐avoiding permutation. Furthermore, the Bruhat order induces a well‐defined order on . Finally, we show that any section of the quotient gives an (often novel) basis for .

Rational noncrossing Coxeter–Catalan combinatorics

AbstractWe 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.

On clusters and exceptional sets

In this paper, we first study clusters in type [Formula: see text] by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type [Formula: see text]. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type [Formula: see text] quivers with straight orientation and using this to count the number of families of exceptional sets of type [Formula: see text] with straight orientation.

Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in [Formula: see text] starting at the origin made of steps [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], where two down-steps cannot be consecutive. We enumerate MAP and their prefixes avoiding peaks (respectively, valleys, respectively, double rise) according to the length, the type of the last step, and the height of its end-point. We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzkin paths.

Certain topological properties for almost convergent Catalan-Motzkin sequence space

The purpose of this paper is to introduce a new almost convergent sequence space by means of a matrix involving Catalan and Motzkin numbers. It is demonstrated that this novel sequence space and the space of all almost convergent sequences are linearly isomorphic and the $\beta$-dual of this new space is deduced. In addition, by defining Catalan-Motzkin core of a complex-valued sequence, certain inclusion relations are derived for it.

Calibrated representations of the double Dyck path algebra

AbstractThe double Dyck path algebra $$\mathbb {A}_{q,t}$$ A q , t and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra $$\mathbb {B}_{q,t}$$ B q , t was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra $$\mathbb {B}_{q,t}$$ B q , t . We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces.

Noncrossing Partition Lattices from Planar Configurations

AbstractThe lattice of noncrossing partitions is well-known for its wide variety of combinatorial appearances and properties. For example, the lattice is rank-symmetric and enumerated by the Catalan numbers. In this article, we introduce a large family of new noncrossing partition lattices with both of these properties, each parametrized by a configuration of n points in the plane.

Explicit generating functions for the sum of the areas under Dyck and Motzkin paths (and for their powers)

Explicit generating functions for the sum of the areas under Dyck and Motzkin paths (and for their powers)

Affine Springer Fibers and Generalized Haiman Ideals (with an Appendix by Eugene Gorsky and Joshua P. Turner)

Abstract We compute the Borel–Moore homology of unramified affine Springer fibers for $\textrm{Gr}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators, and relations, and compare them to generalized $(q,t)$-Catalan numbers. We also compare the homology to the Khovanov–Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case.

Dyck Paths Enumerated by the Q-bonacci Numbers

Dyck Paths Enumerated by the Q-bonacci Numbers

Bijections between Variants of Dyck Paths and Integer Compositions

Bijections between Variants of Dyck Paths and Integer Compositions

Progressive and Rushed Dyck Paths

Progressive and Rushed Dyck Paths

Higher rank (𝑞,𝑡)-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem

We introduce the higher rank ( q , t ) (q,t) -Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a d i n v \mathtt {dinv} statistic on rank r r semistandard ( m , n ) (m,n) -parking functions and prove c o d i n v \mathtt {codinv} counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.

On individual leaf depths of trees

We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work of Panholzer and Prodinger, we apply this method to study the depth distributions of individual nodes or leaves in rooted binary trees, plane trees, noncrossing trees, and increasing trees; the height distributions of individual vertices and individual steps in Dyck paths; and the number of diagonals separating two fixed sides of a convex polygon in a triangulation or dissection. We obtain both exact and asymptotic results, which sometimes refine known formulas or provide combinatorial proofs of results from the probability literature.

2 × 2 minors of recursive matrices related to generalized Dyck paths

A Dyck path of semilength n is a path from (0,0) to (2n,0), such that the only allowed steps are U=(1,1) and D=(1,−1). The constraint on a Dyck path is that it never falls below the x-axis. In this paper, we give the concept of (a1,b1;a2,b2)-Dyck paths, which generates the recursive matrix corresponding to π=(a1,a2,a1,a2,⋯), σ=(0,0,0,0,⋯) and τ=(b1,b2,b1,b2,⋯). Some new combinatorial identities related to 2×2 minors and 2×2 permanents of the Dπ,σ,τ are investigated. When the weighted parameters (a1,b1;a2,b2) are specialized, several interesting identities are obtained about Catalan numbers and other combinatorial sequences. Based on Motzkin paths, alternating sums of 2×2 minors of the odd Dyck matrix are studied.

MIN-turns and MAX-turns in k-Dyck paths: A pure generating function approach

k-Dyck paths differ from ordinary Dyck paths by using an up-step of length k. We analyze at which level the path is after the s-th up-step and before the (s+1)-st up-step. In honour of Rainer Kemp who studied a related concept 40 years ago, the terms max-terms and min-terms are used. Results are obtained by an appropriate use of trivariate generating functions; practically no combinatorial arguments are used.

Skew Dyck paths with air pockets

Skew Dyck paths with air pockets