Related papers: Raised $k$-Dyck paths
We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled.
We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the…
A $k$-ranking of a directed graph $G$ is a labeling of the vertex set of $G$ with $k$ positive integers such that every directed path connecting two vertices with the same label includes a vertex with a larger label in between. The rank…
We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic…
We give a simple proof that the the number of Dyck paths of semilength $n$ with $i$ returns to ground level and $j$ peaks is the generalized Narayana number $\frac{i}{n} \binom{n}{j} \binom{n - i - 1}{j - i}$.
We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating…
We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of…
We compute the limiting distribution of height of a random discrete excursion with step sets consisting of one positive step 1 and arbitrary finite set of non-positive integers. The limit law is the supremum of a Brownian excursion. This is…
Motzkin paths are simple yet important combinatorial objects. In this paper, we consider families of Motzkin paths with restrictions on peak heights, valley heights, upward-run lengths, downward-run lengths, and flat-run lengths. This paper…
For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares…
We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb Z,$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$--Dyck paths and…
Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths, in which tiles are no larger than the tiles they cover. These tilings arise in the study of certain statistical physics models and…
We exhibit a bijection between central Delannoy $n$-paths, that is, lattice paths from the origin to $(n,n)$ with steps $E=(1,0), \,N=(0,1),\,D=(1,1)$ and the lattice paths from the origin to $(n+1,n)$ where the only restriction on the…
We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality…
Let $a,b$ be fixed positive coprime integers. For a positive integer $g$, write $W_k(g)$ for the set of lattice paths from the startpoint $(0,0)$ to the endpoint $(ga,gb)$ with steps restricted to $\{(1,0), (0,1)\}$, having exactly $k$…
We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Fa\`a di Bruno formula,…
Dyck paths with air pockets are obtained from ordinary Dyck paths by compressing maximal runs of down-steps into giant down-steps of arbitrary size. Using the kernel method, we consider partial Dyck paths with air pockets, both, from left…
We consider intersections of n diagonal forms of degrees k 1 < $\bullet$ $\bullet$ $\bullet$ < kn, and we prove an asymptotic formula for the number of rational points of bounded height on these varieties. The proof uses the…
The $k$-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed $k$ when restricted to the class of planar…
We count algebraic numbers of fixed degree $d$ and fixed (absolute multiplicative Weil) height $\mathcal{H}$ with precisely $k$ conjugates that lie inside the open unit disk. We also count the number of values up to $\mathcal{H}$ that the…