Lattice paths inside a table: Rows and columns linear combinations

TL;DR

This paper studies lattice paths within a grid, deriving linear recurrence relations for counting paths and providing formulas for column sums using operator-based methods.

Contribution

It introduces a detailed description of minimal linear recurrences for lattice path counts in tables, advancing combinatorial enumeration techniques.

Findings

Derived linear recurrences for row and column counts

Formulas for total paths from first to last column

Operator methods for analyzing lattice paths

Abstract

A lattice path inside the $m\times n$ table $T$ is a sequence $\nu_1,\ldots,\nu_k$ of cells such that $\nu_{j+1}-\nu_j\in\{(1,-1),(1,0),(1,1)\}$ for all $j=1,\ldots,k-1$. The number of lattice paths in $T$ from the first column to the $(x,y)$-cell is written into that cell. We present a precise description of the minimal linear recurrences among rows, columns, and columns sums. As a result, we obtain several formulas for the number of all lattice paths from the first column to the last column of $T$, that is, the $n^{th}$ column sum. Our methods are based on three classes of operators, which will also be studied independently.