Related papers: Paired patterns in lattice paths

A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has…

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…

Let ${\cal PO}_n$ be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that there exist bijections between the set of certain lattice paths in the Cartesian plane that start at $(0,0)$, end at…

A linkage in a graph $G$ of size $k$ is a subgraph $L$ of $G$ whose connected components are $k$ paths. The pattern of a linkage of size $k$ is the set of $k$ pairs formed by the endpoints of these paths. A consequence of the Unique Linkage…

Matching Logic is a framework for specifying programming language semantics and reasoning about programs. Its formulas are called patterns and are built with variables, symbols, connectives and quantifiers. A pattern is a combination of…

In this paper, we study pattern matching in the set F_{n,k} of fillings of the k x n rectangle with the integers 1,...,kn such that the elements in any column increase from bottom to top. Let P be a column strict tableau of shape 2^k. We…

We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by north-east lattice paths going from the origin to a point (k,n). The numbers from 1 and n…

It is known when we call a poset P, a $\mathcal{P}$-chain permutational poset, given a subset of permutations $\mathcal{P}$ of the symmetric group $S_{n}$. In this work, we use the same idea to study subsets of words of length $n$, that are…

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…

We derive a series of results on random walks on a d-dimensional hypercubic lattice (lattice paths). We introduce the notions of terse and simple paths corresponding to the path having no backtracking parts (spikes). These paths label…

Pattern matching is a widely used technique in functional languages, especially those in the ML and Haskell traditions, where it is at the core of the semantics. In languages in the Lisp tradition, in contrast, pattern matching it typically…

In this note, we discuss planar lattices generated by their atoms. We prove that if $L$ is a planar lattice generated by $n$ atoms, then both the left and the right boundaries of $L$ have at most $n+1$ elements. On the other hand, $L$ can…

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all…

The set of Schr\"oder words (Schr\"oder language) is endowed with a natural partial order, which can be conveniently described by interpreting Schr\"oder words as lattice paths. The resulting poset is called the Schr\"oder pattern poset. We…

We call an interval $[x,y]$ in a poset {\em small} if $y$ is the join of some elements covering $x$. In this paper, we study the chains of paths from a given arbitrary (binary) path $P$ to the maximum path having only small intervals. More…

We solve two problems regarding the enumeration of lattice paths in $\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index, defined as the sum of the positions of the valleys, and to the number of certain crossings.…

In this article we introduce the study of the number of pairs of non-comparable elements in a distributive lattice $\L$. We give several tight lower and upper bounds for the number and give as an application the lattices precisely for which…

We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \geq 0, y \geq 0$ of the square-lattice with unit steps in the…

Path pairs are a modification of parallelogram polyominoes that provide yet another combinatorial interpretation of the Catalan numbers. More generally, the number of path pairs of length $n$ and distance $\delta$ corresponds to the…

We describe two general mechanisms for producing pairing bijections (bijective functions defined from N x N to N). The first mechanism, using n-adic valuations results in parameterized algorithms generating a countable family of distinct…