On Lattice Paths with Several Diagonal Steps

Abstract

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.

Lattice Paths with Diagonal Steps

“…2.8] as can be seen by summing (8) over all k and then over all r. Similarly closed expressions can be obtained for Q(m, n) and Q'(m, n) defined by Rohatgi in [10] for all cases m >_n . 3…”

On a Generalization of One Dimensional Random Walk with a Partially Reflecting Barrier

“…We state below a theorem from [4] in slightly different notations which shall be needed in the subsequent sections. 3.…”

Weighted lattice paths

“…We remark that (2.2) could be obtained directly from equation (1) in [6]. It is also possible to give a simple combinatorial proof of (2.2).…”

“…[1]). The enumeration of the paths which must remain below the line y = ax + b has been obtained by Lyness [4], Mohanty and Narayana [7], and Carlitz, Roselle, and Scoville [2]* Recently several authors [6], [8], and [10] have considered the same problem when diagonal steps are allowed in addition to the usual horizontal and vertical steps, and Stanton and Cowan [11] studied the resulting numbers in a different connection.…”

“…In [6] Mohanty and Handa enumerate the unrestricted lattice paths from (0, 0) to (m, n) where μ diagonal steps are allowed at each position. They also determine the number of such paths which must remain below the line x = ay, and they indicate that a formula can be obtained for the number of paths which must remain below x = ay -6, where a and b are nonnegative integers.…”