A note on partitions in the image of pre$_2$

Technical Summary: A Note on Partitions in the Image of pre2

Problem Statement
This paper addresses a question recently posed by Devnani and Eyyunni regarding the function $pre_2(n)$, which counts the number of integer partitions of $n$ that lie in the image of the map $pre_2$. The map $pre_2$ is defined on a partition $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_\ell)$ by generating a new partition whose parts are all products of $k=2$ distinct parts of $\lambda$. Specifically, the authors investigate whether there exists an integer $n \ge 1$ such that exactly one partition of $n$ is in the image of $pre_2$ (i.e., $pre_2(n) = 1$).

Methodology
The authors employ a constructive approach to demonstrate the existence of multiple partitions in the image of $pre_2$ for sufficiently large $n$.

1. Baseline Observation: The paper first establishes that the partition $(n)$ is always in the image of $pre_2$ because $pre_2(n, 1) = (n)$. Thus, $pre_2(n) \ge 1$ for all $n \ge 1$.
2. Constructive Strategy: To prove $pre_2(n) \ge 2$ for $n \ge 5$, the authors construct specific partitions $\lambda$ with at least three parts such that the sum of the parts of $pre_2(\lambda)$ equals $n$. Since $pre_2(\lambda)$ for a partition with $\ge 3$ parts yields at least $\binom{3}{2} = 3$ parts, the resulting partition cannot be the single-part partition $(n)$. This guarantees a second distinct element in the image.
3. Case Analysis: The proof covers all integers $n \ge 5$ by dividing them into five exhaustive cases based on parity and modular arithmetic (specifically modulo 3 and 4). The constructions utilize partitions of a simple shape, typically consisting of one "large" part $a$ and several small parts (1s or 2s). The sum of the resulting $pre_2(\lambda)$ is calculated using the formula for pairwise products:
   • For $\lambda = (a, 1, \dots, 1)$ with $m-1$ ones, the sum is $a(m-1) + \binom{m-1}{2}$.
   • For $\lambda = (a, 2, 2)$, the sum is $4a + 4$.

Key Contributions and Results
The primary contribution of the paper is the complete resolution of the question posed by Devnani and Eyyunni. The main result is formalized in Theorem 1:

• $pre_2(n) = 1$ if and only if $n \in \{1, 2, 4\}$.
• For every $n \ge 5$, $pre_2(n) \ge 2$.

The authors provide explicit constructions for the five cases covering $n \ge 5$:

• Case 1 (Odd $n \ge 5$): Uses $\lambda = (\frac{n-1}{2}, 1, 1)$.
• Case 2 (Even $n$, $3|n$, $n \ge 6$): Uses $\lambda = (\frac{n}{3}-1, 1, 1, 1)$.
• Case 3 (Even $n$, $3 \nmid n$, $4|n$, $n \ge 12$): Uses $\lambda = (\frac{n-4}{4}, 2, 2)$.
• Case 4 (Even $n$, $3 \nmid n$, $n \equiv 2 \pmod 4$, $n \ge 10$): Uses $\lambda = (\frac{n-6}{4}, 1, 1, 1, 1)$.
• Case 5 ($n=8$): Uses $\lambda = (2, 2, 1)$, yielding $pre_2(2, 2, 1) = (4, 2, 2)$.

Additionally, the paper analyzes small values ($n \le 4$) to confirm that no partitions with three or more parts sum to 1, 2, or 4 under the $pre_2$ map, thereby validating the exception set $\{1, 2, 4\}$. The authors also note a computational verification that $pre_2(n) \ge 2$ holds for all $5 \le n \le 10,000$.

Significance and Scope
The paper modestly claims to provide a definitive answer to a specific open problem in the theory of integer partitions and elementary symmetric polynomials. It does not propose new applications or asymptotic formulas. Instead, it highlights that the constructions used rely on partitions of a "simple shape" (one large part and small parts). The authors suggest that while their work settles the existence question, it leaves open the possibility of obtaining sharper lower bounds for $pre_2(n)$ or finding an asymptotic formula through more varied constructions. The work serves as a precise characterization of the image size of the $pre_2$ map for small and large integers.