However, due to the complexity and size, we apply a known combinatorial method: **counting valid permutations with forbidden adjacency** using inclusion-exclusion or generating functions — but a more efficient way is to use **placement algorithms**.

Tue, 21 Apr 2026 08:16:00 +0000

Counting Valid Permutations With Forbidden Adjacency: Why Placement Algorithms Outperform Inclusion-Exclusion and Generating Functions When tasked with counting valid permutations under complex constraints—especially those involving forbidden adjacents—combinatorial methods like inclusion-exclusion and generating functions often come to mind. While mathematically elegant, these approaches can become cumbersome as problem size increases due to exponential complexity and overlapping inclusion-exclusion terms. A far more efficient and scalable alternative lies in placement algorithms, which directly model valid object placements while avoiding forbidden adjacents—bridging theory and computation with practical insight. ### The Challenge of Forbidden Adjacency in Permutations Consider a problem where you seek to arrange a sequence of distinct elements (e.g., numbers, symbols, or items) such that certain pairs are not adjacent. For example, counting permutations of {1, 2, 3, 4} where 2 and 3 never sit next to each other. This is a classic forbidden adjacency problem, often framed as counting inclusion-valid permutations under adjacency restrictions.