Related papers: Refinements of Lattice paths with flaws

It is a longstanding open problem to find a bijection exchanging area and bounce statistics on Dyck paths. We settle this problem for an exponentially large subset of Dyck paths via an explicit bijection. Moreover, we prove that this…

The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph -- due independently to Kasteleyn and to Temperley and Fisher -- consists of changing the sign of…

We interpret walks in the first quadrant with steps {(1,1),(1,0),(-1,0), (-1,-1)} as a generalization of Dyck words with two sets of letters. Using this language, we give a formal expression for the number of walks in the steps above…

We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon,…

Lattice paths called $\ell$-Schr\"oder paths are introduced. They are paths on the upper half-plane consisting of $\ell+2$ types of steps: $(i,\ell-i)$ for $i=0,\ldots,\ell$, and $(1,-1)$. Those paths generalize Schr\"oder paths and some…

The path integral of a quantum system with an exact symmetry can be written as a sum of functional integrals each giving the contribution from quantum states with definite symmetry properties. We propose a strategy to compute each of them,…

We introduce a subfamily of skew Dyck paths called box paths and show that they are in bijection with pairs of ternary trees, confirming an observation stated previously on the On-Line Encyclopedia of Integer Sequences. More generally, we…

The $k$-deck of a graph is its multiset of induced subgraphs on $k$ vertices. We prove that $n$-vertex graphs with maximum degree $2$ have the same $k$-decks if each cycle has at least $k+1$ vertices, each path component has at least $k-1$…

In 1967, Schmidt wrote a seminal paper [10] on heights of subspaces of R n or C n defined over a number field K, and diophantine approximation problems. The going-down Theorem -- one of the main theorems he proved in his paper -- remains…

There is a strikingly simple classical formula for the number of lattice paths avoiding the line x = ky when k is a positive integer. We show that the natural generalization of this simple formula continues to hold when the line x = ky is…

We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m)…

The concepts of symmetric and asymmetric peaks in Dyck paths were introduced by Fl\'{o}rez and Ram\'{\i}rez, who counted the total number of such peaks over all Dyck paths of a given length. Elizalde generalized their results by giving…

We introduce a new type of lattice path, called brick-wall lattice path, and we derive a formula which counts the number of paths on these lattices imposing certain restrictions on the Cartesian plane. Connections to the Fibonacci sequence,…

We reduce the problem of counting self-avoiding walks in the square lattice to a problem of counting the number of integral points in multidimensional domains. We obtain an asymptotic estimate of the number of self-avoiding walks of length…

We provide new interpretations for a subset of Raney numbers, involving threshold sequences and Motzkin-like paths with long up and down steps. Given three integers n, k, l such that n >= 1, k >= 2 and 0 <= l <= k-2, a (k,l)-threshold…

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…

Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K_n such that the union of any two paths from the family contains a not…

Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run…

Catalan numbers arise in many enumerative contexts as the counting sequence of combinatorial structures. In this work, we consider natural Markov chains on some of the realizations of the Catalan sequence. While our main result is in…

The vicious random walker problem on a one dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the…