Related papers: Generating functions of lattice paths
A rook path is a path on lattice points in the plane in which any proper horizontal step to the right or vertical step north is allowed. If, in addition, one allow bishop steps, that is, proper diagonal steps of slope 1, then one has queen…
We investigate the definability (reducts) lattice of the order of integers and describe a sublattice generated by relations 'between', 'cycle', 'separation', 'neighbor', '1-codirection', 'order' and equality'. Some open questions are…
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also…
We compute generating functions of the set of directed lattice paths starting from the origin and avoiding a periodic set of even point on OX = "time"-axis. As an application we prove a combinatorial identity proposed by P. Hajnal and G.V.…
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,…
We introduce a method for describing Riordan matrices via recurrence relations along their diagonals. This provides a new structural description that complements the classical row-wise and column-wise constructions via the A-sequence. As an…
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 study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
The exponential generating function of ordinary generating functions of diagonal sequences of general Sheffer triangles is computed by an application of Lagrange's theorem. For the special Jabotinsky type this is already known. An analogous…
We begin our analysis with the study of two collections of lattice paths in the plane, denoted $\mathcal{D}_{[n,i,j]}$ and $\mathcal{P}_{[n,i,j]}$. These paths consist of sequences of $n$ steps, where each step allows movement in three…
We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice, and use this cyclic parameterization to count planar well-rounded…
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…
We enumerate the number of monotonic lattice paths starting at $(0,0)$ and terminating at $(m,n)$ in which $l$ of the first $k$ steps lie below the line $y=x\ (0\leq k\leq m\leq n)$. These closed formulas consist of terms which are a…
Osculating paths are sets of directed lattice paths which are not allowed to cross each other or have common edges, but are allowed to have common vertices. In this work we derive a constant term formula for the number of such lattice paths…
By introducing a kind of special functions namely exponent-like function, cosine-like function and sine-like function, we obtain explicitly the basic structures of solutions of initial value problem at the original point for this kind of…
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…
Generating functions for plane overpartitions are obtained using various methods such as nonintersecting paths, RSK type algorithms and symmetric functions. We extend some of the generating functions to cylindric partitions. Also, we show…
We discuss the structure of positive definite kernels in terms of operator models. In particular, we introduce two models, one of Hessenberg type and another one that we call near triangular. These models produce parametrizations of the…
Lattice paths in the quarter plane have led to a large and varied set of results in recent years. One major project has been the classification of step sets according to the properties of the corresponding generating functions, and this has…
We translate the concept of succession rule and the ECO method into matrix notation, introducing the concept of a production matrix. This allows us to combine our method with other enumeration techniques using matrices, such as the method…