Counting lattice paths taking steps in infinitely many directions under special access restrictions

On the Enumeration of Non-Crossing Pairings of Well-Balanced Binary Strings

“…, x n ), where 0 ≤ x i ≤ x and x i is the x-coordinate of the i-th north step. For example, the path EENENNEE is coded by (2,3,3). In other words, the i-th north step of the path has coordinates (x i , i − 1) → (x i , i).…”

“…One advantage of equating a new combinatorial structure to lattice paths is that lattice paths are a classical subject in combinatorial theory, whose enumeration can been systematically approached by symbolic computation and umbral calculus, see, for example, the classical book by Mohanty [6] and the extensive work done by H. Niederhausen and his collaborators [7,8,9,10,11,2,3]. In particular, it is known that lattice paths within general boundaries can by computed by a determinant formula.…”

A history and a survey of lattice path enumeration

“…In Catalan traffic at the beach, Niederhausen (2002) composes the two bijections (reflection and reverse) to give explicit solutions to path counts with S ¼ f/1; 0S; /0; 1Sg above boundaries of rational slope. The papers by Humphreys and Niederhausen (2000, 2004a, 2004b explicitly solve lattice path problems above the horizontal axis and a line with integer slope. Loehr (2003) names f/1; 0S; /0; 1SgÀpaths above a line y= ax+ b trapezoidal lattice paths y when ba0.…”

“…Their example finds a generating function solution for the number of paths with step set S ¼ f/1; 0S; /1; 1S; /1; 2Sg staying below the main diagonal with privileged step set P ¼ f/1; 2Sg. Humphreys and Niederhausen (2000, 2004a, 2004b use finite operator calculus to find explicit solutions to lattice path problems that explore various privileged access step sets for paths that stay above the line y= ax+b where a; b 2 Z.…”