Geometric interpretation of OEIS A078374 using Geometric Composition Algebra (GCA)

This paper establishes a geometric interpretation of OEIS sequence A078374 ("Number of partitions of n into distinct and relatively prime parts") within the Geometric Composition Algebra (GCA) framework. We prove that A078374 counts primitive distinct configurations—geometric objects consisting of horizontal line segments with no repeated lengths and greatest common divisor equal to 1. The main result provides a bijection between partitions satisfying the distinctness and coprimality constraints (classical combinatorial definition) and primitive distinct configurations in GCA (geometric definition). This correspondence is verified computationally for n ≤ 100, with perfect agreement across all tested values. Unlike the related sequence A000837 (which counts all primitive configurations and admits Möbius inversion), A078374 lacks a closed formula. Our geometric perspective suggests new approaches by studying distinctness as a structural constraint on "irregular staircases" within GCA's homothetic orbit theory. This work demonstrates GCA's capacity to naturally encode number-theoretic structures through geometric objects, offering alternative characterizations of combinatorial sequences independent of classical partition-theoretic methods.