Extending Andrews and Newman's refinement of the crank-mex theorem

Problem Statement
The paper addresses the refinement of the "crank–mex theorem," a result established by Andrews and D. Newman (2020) and Hopkins and Sellers (2020). The original theorem states that the number of integer partitions of $n$ with a nonnegative crank ($M_{\ge 0}(n)$) equals the number of partitions of $n$ with an odd minimal excludant (mex) ($X_o(n)$). Andrews and M. Newman recently refined this equality by tracking the number of parts greater than one, denoted by the statistic $\beta(\lambda)$. They proved analytically that $x_o(n, k) = m_{\ge 0}(n, k)$ but requested a combinatorial proof.

The current work extends this line of inquiry to a complementary set of partition statistics. Specifically, it investigates the equinumerosity between:

1. Partitions with an even mex ($X_e(n)$).
2. Partitions with a fixed point (where $\lambda_i = i$ for some $i$).
3. Partitions with a negative crank ($M_{< 0}(n)$).
4. Partitions with a positive crank ($M_{> 0}(n)$).

The authors aim to establish and refine these relationships by tracking the number of parts greater than one ($k$) and, for the fixed-point case, the specific count of such parts.

Methodology
The paper employs two distinct methodologies to prove the main theorem:

1. Analytic Proof: The authors utilize $q$-series and generating functions. They define generating functions for each of the four partition sets, introducing a variable $z$ to track the number of parts greater than one and $q$ to track the sum of the parts. For the crank-related sets, they utilize the known crank generating function derived by Andrews and Garvan (1988). Through algebraic manipulations, including the use of Heine transforms and $q$-binomial theorems, they demonstrate that the generating functions for these sets are identical (or related by a simple factor of $z$), thereby proving the equinumerosity.
2. Combinatorial Proof: The authors construct explicit bijections between the sets.
   • Even Mex $\leftrightarrow$ Fixed Points: They adapt a generalized bijection by Konan (2023) that connects partitions with odd mex to partitions without fixed points. They modify this to connect even mex partitions to partitions with a fixed point, carefully analyzing how the number of parts greater than one changes during the iterative mapping.
   • Fixed Points $\leftrightarrow$ Negative Crank: They define a bijection based on the relationship between the fixed point index, the number of 1s ($\omega(\lambda)$), and the crank. Lemmas are established to show that partitions with a fixed point and negative crank satisfy specific bounds on $\omega(\lambda)$. The map involves replacing a fixed point $i$ with $i$ ones (or vice versa), adjusting the count of parts greater than one accordingly.
   • Negative Crank $\leftrightarrow$ Positive Crank: A direct bijection is constructed between partitions with negative crank and those with positive crank. This map involves shifting parts and the number of 1s to invert the sign of the crank while increasing the number of parts greater than one by exactly one.

Key Contributions and Results
The primary result is Theorem 1, which establishes that for positive integers $n$ and $k$, the following four sets are equinumerous:

• (a) Partitions of $n$ with even mex and $k$ parts greater than 1.
• (b) Partitions of $n$ with a fixed point and $k+1$ parts greater than 1.
• (c) Partitions of $n$ with negative crank and $k$ parts greater than 1.
• (d) Partitions of $n$ with positive crank and $k+1$ parts greater than 1.

Mathematically, this is expressed as:
$$x_e(n, k) = f(n, k+1) = m_{< 0}(n, k) = m_{> 0}(n, k+1)$$

The paper provides both analytic and combinatorial proofs for this theorem. The combinatorial proofs rely on a chain of bijections:
$$X_e(n, k) \xrightarrow{\text{Konan's map}} (1) \times G_1(n-1, k) \xrightarrow{\text{Fixed point map}} F(n, k+1) \xrightarrow{\text{Crank map}} M_{< 0}(n, k) \xrightarrow{\text{Crank map}} M_{> 0}(n, k+1)$$

The authors also address the specific case where $k=0$, noting a minor exception regarding the partition $(1^n)$ and introducing the notation $F^*(n, k+1)$ to handle this boundary condition.

Significance and Limitations
The paper claims to successfully extend the refinement of the crank–mex theorem to the complementary sets involving even mex, fixed points, and signed cranks. By providing both analytic and combinatorial proofs, the work offers a deeper structural understanding of these partition statistics.

However, the authors are modest regarding the resolution of the open problem posed by Andrews and Newman. They explicitly state that their results do not provide a combinatorial proof for the refinement $x_o(n, k) = m_{\ge 0}(n, k)$. They identify two "insurmountable impediments" to refining Konan's existing bijections for this specific case:

1. There is no bijection between $X_o(n)$ and the set of partitions without fixed points ($G(n)$) that preserves the number of parts greater than one, as the distributions of $\beta$ values differ fundamentally between these sets.
2. There is no bijection between $M_{\le 0}(n, k)$ and $M_{\ge 0}(n, k+1)$ that preserves the necessary statistics, even when accounting for the parity of crank-zero partitions.

Consequently, the paper concludes that satisfying the request for a combinatorial proof of the odd-mex refinement would likely require a direct connection between $X_o(n)$ and $M_{\ge 0}(n)$ or the discovery of entirely new equinumerous sets, rather than an extension of the current bijections.