Related papers: Catalan pairs and Fishburn triples
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…
We show that matchings avoiding certain partial patterns are counted by the 3-Catalan numbers. We give a characterization of 12312-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a…
In this paper, we introduce a metric on the set of pairs of coprime natural numbers. We explicitly construct a quasi-isometric embedding from the set of natural numbers into this metric space via Fibonacci numbers.
The purpose of this paper is twofold. First we answer to a question asked by Steingrimsson and Williams about certain permutation tableaux: we construct a bijection between binary trees and the so-called Catalan tableaux. These tableaux are…
The theory of quasirandomness has greatly expanded from its inaugural graph theoretical setting to several different combinatorial objects such as hypergraphs, tournaments, permutations, etc. However, these quasirandomness variants have…
We introduce near triple arrays as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near…
We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2,…
Towards the study of the Kashiwara B(infinity) crystal, sets H^t of functions were introduced given by equivalence classes of unordered partitions satisfying certain boundary conditions. Here it is shown that H^t is a Catalan set of order…
The notion of $(a,b)$-cores is closely related to rational $(a,b)$ Dyck paths due to Anderson's bijection, and thus the number of $(a,a+1)$-cores is given by the Catalan number $C_a$. Recent research shows that $(a,a+1)$ cores with distinct…
In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are…
We define q-Catalan bases which are a generalization of the q-polynomials z^n(z,q)_n. The determination of their dual bases involves some q-power series termed dual coefficients. We show how these dual coefficients occur in the solution of…
This is a largely expository paper in which we discuss various sets having a Catalan number of objects and some well-known bijections between these sets presented in a new and hopefully interesting way. We call these concepts "bookshelf"…
In a bicategory of spans (an example of a 'generic bicategory') the factorization of a span (s,t) as the span (s,1) followed by (1,t) satisfies a simple universal property with respect to all factorizations in terms of the generic…
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…
In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete…
In this short note, we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers.
In this note, we present some basic properties of $q$-Fibonacci numbers and their relationship to the $q$-golden ratio and Catalan numbers. We then use this relationship to give a short proof of a combinatorial identity.
Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{Set}$ from the category with two parallel arrows. Such functors are…
The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later on translated into a purely combinatorial…
A Catalan magma is a unique factorisation normed magma with only one irreducible element. The \val partitions the base set into subsets enumerated by Catalan numbers. The primary theorem characterises the conditions which a set a with…