Two-color partitions with evens in one color

TL;DR

This work analyzes two-color integer partitions in which even parts must appear in blue, introducing counts $F(n)$, $F_0(n)$, $F_1(n)$, $F_2(n)$, and $F_3(n)$ for various parity/color constraints. It derives explicit generating functions for $F_0$ and $F_1$ using $q$-series techniques (Euler identities and Jacobi's triple product) and connects $F_2$ and $F_3$ to minimal excludant partitions in the blue subset, yielding exact product-sum representations. The paper also reveals rich connections to overpartitions, providing expressions like $F_0(n)=rac{ar p(n)+ar p(n/2)}{2}$ and $F_1(n)=rac{ar p(n)-ar p(n/2)}{2}$, and shows that $F(n)$ equals the overpartition count $ar p(n)$. Together, these results give new identities and interpretations in the realm of $q$-series and partition theory, with potential bijective proofs and further identities proposed as directions for future work.

Abstract

We consider sequences counting integer partitions in two colors (red and blue) in which the even parts occur only in blue color. We focus on subsequences defined by constraints on the parity and color of the summands. We establish formulas for our sequences and deduce identities of integer partitions.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Theorem 1
• Corollary 1
• Theorem 2
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• proof