Related papers: Lattice paths inside a table, II
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,…
Recently, Brualdi and Cao studied $I_k$-avoiding $(0,1)$-matrices by decomposing them into zigzag paths and proved that the maximum number of $1$'s in such a matrix is given by an exact formula. We further study the structure of maximal…
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. L. Shapiro [9] found the Chung-Feller properties for the Motzkin paths.…
We consider lattice walks in $\R^k$ confined to the region $0<x_1<x_2...<x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using…
Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pockets by adding some horizontal steps with certain conditions. In this paper, we introduce two generalizations. The first one consists of lattice…
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…
Let $v$ be a grid path made of north and east steps. The lattice $\rm{T{\scriptsize AM}}(v)$, based on all grid paths weakly above $v$ and sharing the same endpoints as $v$, was introduced by Pr\'eville-Ratelle and Viennot (2014) and…
We study the expected distance of short uniform random walks in arbitrary dimensions with unit steps in random directions. It is known that for dimensions $d=2$ and $d=4$, all the moments of an $m$-step walk are integer. While for $d=2$,…
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…
We present a complete solution to the so-called tennis ball problem, which is equivalent to counting lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit…
We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block $P_j$ take on $c_j$ colors and count all…
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…
Let $I_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n+1}x_{n+2}\cdots x_{n+m})$ be the $m$-path ideal of a path of length $n + m-1$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{n+m}]$. We compute all the graded Betti…
Similar to how standard Young tableaux represent paths in the Young lattice, Latin rectangles may be use to enumerate paths in the poset of semi-magic squares with entries zero or one. The symmetries associated to determinant preserve this…
For each $p>0$ we define by recurrence a triangle $T^p(n,k)$ whose rows sum to the Fuss-Catalan numbers $ \frac{1}{p n+1}\binom{pn+1}{n}$, generalizing the known Catalan triangle corresponding to the case $p=2$. (In fact, $T^p(n,k)$ has an…
Using lattice path counting arguments, we reproduce a well known formula for the number of standard Young tableaux. We also produce an interesting new formula for tableaux of height $\leq 3$ using the Fourier methods of Ault and Kicey.
We derive a path counting formula for two-dimensional lattice path model on a plane with filter restrictions. A filter is a line that restricts the path passing it to one of possible directions. Moreover, each path that touches this line is…
In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\`ere, where cover relations…
In this article we will show that for every natural $d$ and $n>1$ there exists a natural number $t$ such that for every $d$-dimensional simplicial complex $\mathcal{T}$ with vertices in $\mathbb{Z}^d$, the number of lattice points in the…
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four…