Lattice Paths

Combinatorics · Axiom Academy

Discovering the combinatorial structure of grid navigation 1. The Problem: Counting Grid Paths A lattice path from point (0,0) to point (m,n) consists of moves on a grid where we can only move right (R) or up (U) . Question: How many distinct paths exist from (0,0) to (m,n)? Every path from (0,0) to (m,n) requires: Each path is a unique arrangement of m R's and n U's. The question becomes: In how many ways can we arrange m identical R's and n identical U's? We need to select m positions for R moves from m+n total positions: Equivalently, we can choose which n positions will be U moves: Let's verify with specific cases: 5. Extensions and Related Problems Restricted Paths: What if certain grid points are blocked? We can use the Inclusion-Exclusion Principle : Total paths minus paths through blocked points. 6. Bijections to Other Counting Problems Lattice paths provide a unified framework for understanding many combinatorial problems:

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