Related papers: Catalan numbers and relations

We present the new combinatorial class of product-coproduct prographs which are planar assemblies of two types of operators: products having two inputs and a single output and coproducts having a single input and two outputs. We show that…

In this paper, we study the combinatorial structures of straight and ordinary m\'enage permutations. Based on these structures, we prove four formulas. The first two formulas define a relationship between the m\'enage numbers and the…

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…

In this note we give a survey about polynomials whose moments are multiples of super Catalan numbers and explore two different kinds of q-analogues.

The lonely singles sequence represents the number of noncrossing partitions of the finite set {1,. .. , n} in which no pair of singletons {i} and {j} can be merged into the pair {i, j} so that the partition stays noncrossing. The…

We introduce the notion of a Mahonian pair. Consider the set, P^*, of all words having the positive integers as alphabet. Given finite subsets S,T of P^*, we say that (S,T) is a Mahonian pair if the distribution of the major index, maj,…

In this paper, we define four transformations on the classical Catalan triangle $\mathcal{C}=(C_{n,k})_{n\geq k\geq 0}$ with $C_{n,k}=\frac{k+1}{n+1}\binom{2n-k}{n}$. The first three ones are based on the determinant and the forth is…

We introduce the $q,t$-Catalan measures, a sequence of piece-wise polynomial measures on $\mathbb{R}^2$. These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and…

In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem…

The Cartan scheme $\cal X$ of a finite group $G$ with a $(B,N)$-pair is defined to be the coherent configuration associated with the action of $G$ on the right cosets of the Cartan subgroup $B\cap N$ by the right multiplications. It is…

In this paper, we define the concepts of semi-canonical and canonical binary matrix. Strictly mathematical, we prove the correctness of these definitions. We describe and we implement an algorithm for finding all semi-canonical binary…

In this paper we introduce a notion of rational singularities associated to pairs $(X, \ba^t)$ where $X$ is a variety, $\ba$ is an ideal sheaf and $t$ is a nonnegative real number. We prove that most standard results about rational…

This paper establishes an interesting link between $k$th price auctions and Catalan numbers by showing that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a $k$th price auction with…

The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was…

The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the…

We present an exposition on the Fuss--Catalan numbers, which are a generalization of the well known Catalan numbers. The literature on the subject is scattered (especially for the case of multiple independent parameters, as will be…

This paper proposes a new setup for studying pairs of structures. This new framework includes many of the previously studied classes of pairs, such as dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and…

Li et al. give an integral formula for the Catalan-Qi number of the second kind. They show that this integral can be written as a summation with double factorials. In this paper the integral is reduced to a product of the Catalan number and…

I study Hankel determinants of a class of sequences which can be interpreted as generalizations of the Catalan numbers and the central binomial coefficients. They follow a modular pattern with a frequent appearance of zeroes, so that the…

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…