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Counting generalized Dyck paths
Abstract
The Catalan number has a lot of interpretations and one of them is the number
of Dyck paths. A Dyck path is a lattice path from (0,0) to (n,n) which is
below the diagonal line y=x. One way to generalize the definition of Dyck
path is to change the end point of Dyck path, i.e. we define (generalized) Dyck
path to be a lattice path from (0,0) to which is
below the diagonal line , and denote by C(m,n) the number of
Dyck paths from (0,0) to (m,n). In this paper, we give a formula to
calculate C(m,n) for arbitrary m and n.
... On a two-letter alphabet they correspond to well parenthesized expressions and can be interpreted in terms of paths in a square. Among the many possible generalizations, it is natural to consider paths in a rectangle, see for instance Labelle and Yeh [11], and more recently Duchon [5] or Fukukawa [7]. In algebraic combinatorics Dyck paths are related to parking functions and the representation theory of the symmetric group [8]. ...
We consider the problem of counting the set of of Dyck
paths inscribed in a rectangle of size . 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 with a and b non relatively prime, in
terms of Catalan numbers.
... Here we draw two lattice paths as defined above, both with starting point (0, 0) and end point (4,5). The green one is a generalized Dyck path, because it is always above the blue line 4y = 5x, while the red one is not a generalized Dyck path. ...
We classify all solutions (p,q) to the equation p(u)q(u)=p(u+b)q(u+a) where p
and q are complex polynomials in one indeterminate u, and a and b are fixed but
arbitrary complex numbers. This equation is a special case of a system of
equations which ensures that certain algebras defined by generators and
relations are non-trivial. We first give a necessary condition for the
existence of non-trivial solutions to the equation. Then, under this condition,
we use combinatorics of generalized Dyck paths to describe all solutions and a
canonical way to factor each solution into a product of irreducible solutions.
La idea de este trabajo es elaborar generalizaciones de las f ́ormulas obtenidas para la enumeraci ́on de los caminos de Dyck incluidos en un diagrama de Ferrers. Para ello introduciremos un polinomio enumerativo multi-variables indexada por un conjunto de particiones.
We provide a new algebraic grammar for generalized Dyck languages as introduced by Labelle and Yeh (Discrete Math. 82 (1990) 1–6). The study of these languages leads to the particular sublanguages of words without proper factors belonging to the studied language. A random generation scheme is shown for generalized Dyck languages, which leads to some asymptotic results. In the two-letter case, for which the words correspond to ‘rational slope Dyck paths’, more exact and asymptotic enumerative results are obtained, including the asymptotic average area to integer or 32 slope Dyck paths.