[Paper Review] On Directed Lattice Paths With Additional Vertical Steps

Maciej Dziemiańczuk|arXiv (Cornell University)|Oct 20, 2014

Advanced Combinatorial Mathematics12 references3 citations

TL;DR

This paper introduces a combinatorial framework for directed lattice paths that include vertical steps (0,-1) and non-vertical steps (1,k) for integer k. It establishes a bijection between primary paths (with vertical steps) and weighted non-vertical paths, showing that the number of primary paths equals the sum of weights over corresponding weighted paths. A key result is that the expected number of vertical steps in a primary path from (0,0) to (n,-1) equals the number of free paths from (0,0) to (n,0), linking path enumeration to generating functions and known combinatorial sequences like Dyck, Motzkin, and Delannoy paths.

ABSTRACT

The paper is devoted to the study of lattice paths that consist of vertical steps $(0,-1)$ and non-vertical steps $(1,k)$ for some $k\in \mathbb Z$. Two special families of primary and free lattice paths with vertical steps are considered. It is shown that for any family of primary paths there are equinumerous families of proper weighted lattice paths that consist of only non-vertical steps. The relation between primary and free paths is established and some combinatorial and statistical properties are obtained. It is shown that the expected number of vertical steps in a primary path running from $(0,0)$ to $(n,-1)$ is equal to the number of free paths running from $(0,0)$ to $(n,0)$. Enumerative results with generating functions are given. Finally, a few examples of families of paths with vertical steps are presented and related to Łukasiewicz, Motzkin, Dyck and Delannoy paths.

Motivation & Objective

• To study lattice paths that include vertical steps (0,-1) and non-vertical steps (1,k) for k ∈ ℤ.
• To establish a bijection between primary paths (with vertical steps) and weighted non-vertical paths.
• To derive generating functions and closed-form expressions for path enumerations.
• To relate path families to classical combinatorial objects such as Dyck, Motzkin, and Delannoy paths.
• To compute expected numbers of vertical steps in primary paths and relate them to free path counts.

Proposed method

• Define two path families: free paths (F_S(m,n)) and primary paths (P_S(m,n)) from (0,0) to (n,-m), with primary paths constrained to stay on or above the x-axis.
• Construct a weight function w on non-vertical steps such that |P_V(m,n)| = sum over w-weighted paths in P_L(m,n), where L is a transformed step set.
• Use a bijection between primary V-paths and w-weighted primary L-paths to prove equinumerosity.
• Apply the kernel method and generating function techniques to derive closed-form expressions for path counts.
• Transform lattice paths via coordinate mapping (i,j) ↦ (i−j,i) to relate Delannoy paths to paths with steps V, U₁, U₀.
• Derive recurrence and generating function relations for path families using symbolic combinatorics and coefficient extraction.

Experimental results

Research questions

• RQ1 How can lattice paths with vertical steps be bijectively encoded using weighted non-vertical steps?
• RQ2 What is the relationship between the number of vertical steps in primary paths and the count of free paths?
• RQ3 How do generating functions for path families with vertical steps relate to those without?
• RQ4 To what extent can classical paths like Dyck, Motzkin, and Delannoy paths be unified under this framework?
• RQ5 What is the expected number of vertical steps in a primary path from (0,0) to (n,-1), and how does it relate to free path counts?

Key findings

• The expected number of vertical steps in a primary path from (0,0) to (n,-1) is equal to the number of free paths from (0,0) to (n,0), i.e., E[|V|] = |F_V(0,n)|.
• For any family of primary paths with step set V ⊆ {V} ∪ Ω_N containing U_N and V, there exists a corresponding weighted path family over L = (V ∖ {V}) ∪ {D₁, U₀, ..., U_N} such that |P_V(m,n)| = ∑_{π∈P_L(m,n)} w(π).
• The number of primary paths |P_E(1,n)| for E = {V, U₁, U₀} equals the large Schröder number S(n), and is given by |P_E(1,n)| = (D(n,n+1) - D(n,n))/n.
• The number of Delannoy paths from (0,0) to (n,k) is D(n,k) = ∑_{j=0}^k binom(n,j) binom(n+k-j,n), and D(n,n) is the central Delannoy number.
• The generating function for primary paths with step set D = {V, U₁, U₀} satisfies P_1(x) = (1 - √(1 - 4x - 4x²))/(2x), matching OEIS A025227.
• The number of free paths |F_E(1,n)| is given by ∑_{j=0}^n ∑_{k=0}^{⌊(Nj+1)/(N+1)⌋} (-1)^{n-j} / j * binom(j,k) * binom((N+1)(j-k), j-1), with N=1 yielding OEIS A179191.