Related papers: Generating functions of lattice paths
This paper concentrates on the set $\mathcal{V}_n$ of weighted Dyck paths of length $2n$ with special restrictions on the level of valleys. We first give its explicit formula of the counting generating function in terms of certain weight…
There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide…
The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their…
Recent work of the author connected several parking function enumeration problems to enumerations of Catalan paths with respect to certain weight functions that are expressed in terms of the ascent lengths. Motivated by this, we generalise…
We define a so-called square $k$-zig-zag shape as a part of the regular square grid. Considering the shape as a $k$-zig-zag digraph, we give values of its vertices according to the number of the shortest paths from a base vertex. It…
Generalized Dyck paths (or discrete excursions) are one-dimensional paths that take their steps in a given finite set S, start and end at height 0, and remain at a non-negative height. Bousquet-M\'elou showed that the generating function…
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m,r) and that remain in the region bounded by P and Q can be identified with…
In this paper we studied infinite weighted automata and a general methodology to solve a wide variety of classical lattice path counting problems in an uniform way. This counting problems are related to Dyck paths, Motzkin paths and some…
Let a and b be two positive integers. A culminating path is a path of Z^2 that starts from (0,0), consists of steps (1,a) and (1,-b), stays above the x-axis and ends at the highest ordinate it ever reaches. These paths were first…
Lattice paths effectively model phenomena in chemistry, physics and probability theory. Asymptotic enumeration of lattice paths is linked with entropy in the physical systems being modeled. Lattice paths restricted to different regions of…
We classify and construct all line graphs that are $3$-polytopes (planar and $3$-connected). Apart from a few special cases, they are all obtained starting from the medial graphs of cubic (i.e., $3$-regular) $3$-polytopes, by applying two…
We use production matrices to count several classes of geometric graphs. We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of…
$k$-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…
We introduce an algorithm for computing closure systems derived from a family of implications on a set. Semilattices presentations are explored and used in conjunction with the algorithm to compute various types of lattices freely generated…
The direct application of the definition of sorting in lattices is impractical because it leads to an algorithm with exponential complexity. In this paper we present for distributive lattices a recursive formulation to compute the sort of a…
For an integer $p\geq 2$ we construct vertical and horizontal one-pth Riordan arrays from a Riordan array. When $p=2$, one-pth Riordan arrays reduced to well known half Riordan arrays. The generating functions of the $A$-sequences of…
We consider module categories of path algebras of connected acyclic quivers. It is shown in this paper that the set of functorially finite torsion classes form a lattice if and only if the quiver is either Dynkin quiver of type A, D, E, or…
Many widely used network centralities are based on counting walks that meet specific criteria. This paper introduces a systematic framework for walk enumeration using generating functions. We introduce a first-passage decomposition that…
We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, 'jumps' orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with…
Koroljuk gave a summation formula for counting the number of lattice paths from $(0,0)$ to $(m,n)$ with $(1,0), (0,1)$-steps in the plane that stay strictly above the line $y=k(x-d)$, where $k$ and $d$ are positive integers. In this paper…