Catalan Numbers Research Articles

Pattern avoidance in biwords

In this paper, we study pattern avoidance on the set of biwords with no repetitions in each block and prove that the number of those biwords avoiding π‾ is independent of the choice of π∈S3, which extends Knuth's classic result on permutations avoiding π∈S3. We present the proof in both bijective and inductive methods. As applications, we will give new combinatorial interpretations on Catalan, Riordan, and Motzkin numbers via pattern avoidance of biwords. Using the elementary theory of symmetric functions, we investigate the generating functions of biwords avoiding several specific patterns.

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which M\'esz\'aros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize M\'esz\'aros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-M\'esz\'aros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity. Lastly, we study a symmetry of the Morris identity bijectively using the Danilov-Karzanov-Koshevoy triangulation of flow polytopes and a bijection of M\'esz\'aros-Morales-Striker.

Explicit formulas and recurrence relations for generalized Catalan numbers

In this paper, we present an explicit formula and recurrent relation for generalized Catalan numbers, from which we can give corresponding formulas for Schröder numbers, large and small generalized Catalan numbers for the special cases of our results.

Chase-escape with death on trees

Chase-escape is a competitive growth process in which red particles spread to adjacent uncolored sites, while blue particles overtake adjacent red particles. We introduce the variant in which red particles die and describe the phase diagram for the resulting process on infinite d-ary trees. A novel connection to weighted Catalan numbers makes it possible to characterize the critical behavior.

Endomorphisms of upper triangular matrix semirings

We give a description of the class of endomorphisms in the semiring of upper triangular matrices over an additively idempotent semiring S. The endomorphisms α such that where Eij is a matrix unit, is a (0,1)-matrix are discussed. Our focus of investigation concerns the idempotent endomorphisms of this type. In the last theorem we prove that the number of idempotents is the (n + 1)-th Catalan number

Minimal Kinematics: An All k and n Peek into Trop<sup>+</sup>G(k,n)

In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all $k$ and $n$ on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of $n$ points on $\mathbb{CP}^{k-1}$ has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all $k$ generalization of the one recently introduced by Early for $k=2$ and uses a choice of cyclic ordering. We conjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which we have checked analytically through $n=10$ for all $k$. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph ${\rm C}_n^{(1,2,\dots, k-2)}$. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian ${\rm Tr}^+{\rm G}(k,n)$ in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the $90\,608$ planar arrays for ${\rm Tr}^+{\rm G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called planar-basis kinematics which generalizes the one introduced by Cachazo, He and Yuan for $k=2$ and uses the planar basis recently introduced by Early for all $k$. Based on numerical computations through $n=8$ for all $k$, we conjecture that on the planar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For $k=2$ these are the standard Catalan numbers.

Patterns in Shi Tableaux and Dyck Paths

Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type A, the set T $\mathcal {T}$ of Shi tableaux naturally coincides with the set of Dyck paths, for which various notions of patterns have been introduced and studied over the years. In this paper we define a notion of pattern occurrence in T $\mathcal {T}$ which, although it can be regarded as a pattern on Dyck paths, it is motivated by the underlying geometric structure of the tableaux. Our main goal in this work is to study the poset of Shi tableaux defined by pattern-containment. More precisely, we determine explicit formulas for upper and lower covers for each T ∈ T $T\in \mathcal {T}$ , we consider pattern avoidance for the smallest non-trivial tableaux (size 2) and generalize these results to certain tableau of larger size. We conclude with open problems and possible future directions.

The Degree of Symmetry of Lattice Paths

The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We study the behavior of this statistic on Dyck paths and grand Dyck paths, with symmetry described by reflection along a vertical line through their midpoint; partitions, with symmetry given by conjugation; and certain compositions interpreted as bargraphs. We find expressions for the generating functions for these objects with respect to their degree of symmetry, and their semilength or semiperimeter, deducing in most cases that, asymptotically, the degree of symmetry has a Rayleigh or half-normal limiting distribution. The resulting generating functions are often algebraic, with the notable exception of Dyck paths, for which we conjecture that it is D-finite (but not algebraic), based on a functional equation that we obtain using bijections to walks in the plane.

Refined Restricted Inversion Sequences

Recently, the study of patterns in inversion sequences was initiated by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck independently. Motivated by their works and a double Eulerian equidistribution due to Foata (1977), we investigate several classical statistics on restricted inversion sequences that are either known or conjectured to be enumerated by Catalan, Large Schröder, Baxter and Euler numbers. One of the two highlights of our results is a fascinating bijection between 000-avoiding inversion sequences and Simsun permutations, which together with Foata’s V- and S-codes, provide a proof of a restricted double Eulerian equidistribution. The other one is a refinement of a conjecture due to Martinez and Savage that the cardinality of $${\mathbf{I}}_n(\ge ,\ge ,>)$$ is the n-th Baxter number, which is proved via the so-called obstinate kernel method developed by Bousquet-Mélou.

The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

Enumerations of Rational Non-decreasing Dyck Paths with Integer Slope

Enumerations of Rational Non-decreasing Dyck Paths with Integer Slope

Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats

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 special cases includes central binomial coefficients and Catalan numbers as well as relations to enumeration of another family of walks previously studied for which we provide bijection.

Restricted stacks as functions

The stack sort algorithm has been the subject of extensive study over the years. In this paper we explore a generalized version of this algorithm where instead of avoiding a single decrease, the stack avoids a set T of permutations. We let sT denote this map. We classify for which sets T the map sT is bijective. A corollary to this answers a question of Baril, Cerbai, Khalil, and Vajnovszki about stack sort composed with s{σ,τ}, known as the (σ,τ)-machine. This fully classifies for which σ and τ the preimage of the identity under the (σ,τ)-machine is counted by the Catalan numbers. We also prove that the number of preimages of a permutation under the map sT is bounded by the Catalan numbers, with a shift of indices. For T of size 1, we classify exactly when this bound is sharp. We also explore the periodic points and maximum number of preimages of various sT for T containing two length 3 permutations.

Generalizing dendriform algebras: Dyckm-algebras, Rotam-algebras, and Rota–Baxter operators

We review the notion of a [Formula: see text], an algebraic structure introduced recently by López, Préville-Ratelle and Ronco during their work on the splitting of associativity via [Formula: see text]-Dyck paths, and we also introduce Rota[Formula: see text]-algebras: both structures can be considered as generalizations of dendriform structures. We obtain examples of Dyck[Formula: see text]-algebras in terms of planar rooted binary trees equipped with a particular type of Rota–Baxter operator, and we present examples of Rotam-algebras using left averaging morphisms. As an application, we observe that the structures presented here allow us to introduce quite naturally a “non-associative version” of the Kadomtsev–Petviashvili hierarchy.

Preimages under the Queuesort algorithm

Following the footprints of what has been done with the algorithm Stacksort, we investigate the preimages of the map associated with a slightly less well known algorithm, called Queuesort. After having described an equivalent version of Queuesort, we provide a recursive description of the set of all preimages of a given permutation, which can be also translated into a recursive procedure to effectively find such preimages. We then deal with some enumerative issues. More specifically, we investigate the cardinality of the set of preimages of a given permutation, showing that all cardinalities are possible, except for 3. We also give exact enumeration results for the number of permutations having 0,1 and 2 preimages. Finally, we consider the special case of those permutations π whose set of left-to-right maxima is the disjoint union of a prefix and a suffix of π: we determine a closed formula for the number of preimages of such permutations, which involves two different incarnations of ballot numbers, and we show that our formula can be expressed as a linear combination of Catalan numbers.

Binomial sums involving Catalan numbers

Recently, various interesting binomial sums involving Catalan numbers have been derived and computed. Inspired from them, several binomial sums involving Catalan numbers are evaluated in closed forms using a hypergeometric series approach.

Hamiltonian and exclusion statistics approach to discrete forward-moving paths.

We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an alternative, simpler form for its logarithm that makes its polynomial structure explicit.

Action graphs, rooted planar forests, and self-convolutions of the Catalan numbers

We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of trees.

A note on the moments of the Kesten distribution

In this paper, we present two explicit expressions of the moments M2k of the Kesten distribution in terms of Catalan's triangles Tk,i=k−i+1k+1(k+ik) and Shapiro's Catalan triangles Bk,j=jk(2kk−j), respectively. As a corollary, we provide a generalization of the identity, due to Eplett, between the Catalan number Ck=1k+1(2kk) and the triangles Bk,j. As an application, we obtain an asymptotic formula on M2k as k tends to infinity.

Enumeration formulas for standard young tableaux of approximate C shape

This paper focuses on the combinatorial enumeration of the truncated hollow standard Young tableaux of approximate C shape. The enumeration formula of approximately C-shaped SYT, namely the convolution of Catalan numbers, is derived by using the multi-integral method based on the sequence statistical model. Meanwhile, it is found that the first and last rows differ in the number of boxes. In the case where the number of boxes in the last row n is large enough, the number of SYT does not change, especially if m = 1, whether it is closely related to the Catalan number. In addition, we present two main closed formulas for SYTs of angular and “T” shapes.