Overpartitions with repeated smallest non-overlined part

1. Problem Statement and Motivation

The paper extends recent work by Andrews and Bachraoui regarding integer partitions where the smallest part is repeated exactly $k$ times and the remaining parts are distinct. The authors aim to generalize these results to the realm of overpartitions.

• Context: An overpartition is a partition where the first occurrence of a number may be overlined. Overpartitions are a central object in modern partition theory, connecting combinatorics, $q$-series, and modular forms.
• The Core Problem: The authors define specific classes of overpartitions based on the behavior of the smallest non-overlined part (denoted $s(\pi)$).
  • Class 1 ($Spt_k(n)$): Overpartitions of $n$ where the smallest non-overlined part appears exactly $k$ times, and every overlined part is strictly larger than $s(\pi)$.
  • Class 2 ($Spt_k^o(n)$): A subclass of the above where, in addition to the previous conditions, all parts other than $s(\pi)$ are incongruent modulo 2 with $s(\pi)$ (i.e., they have different parity).
• Goal: To derive closed-form generating functions for the cardinalities of these sets ($spt_k(n)$ and $spt_k^o(n)$) and their signed variations ($spt_k'(n)$ and $spt_k'^o(n)$), expressing them as linear combinations of $q$-Pochhammer symbols with rational function coefficients.

2. Methodology

The authors employ a combination of combinatorial generating function manipulation and analytic techniques specific to $q$-series.

• Generating Function Derivation:
  • The authors first establish the generating functions for the defined classes using standard combinatorial arguments.
  • For $spt_k(n)$, the generating function is derived as:
    $$\sum_{n=1}^\infty spt_k(n)q^n = \sum_{n=1}^\infty \frac{q^{kn}(-q^{n+1}; q)_\infty}{(q^{n+1}; q)_\infty}$$
  • Similar expressions are derived for the signed versions and the parity-restricted subclasses.
• Key Analytic Tools:
  • $q$-Binomial Theorem: Used extensively to sum series of the form $\sum \frac{(a;q)_n}{(q;q)_n} z^n$.
  • Andrews-Subbarao-Vidyasagar Formula: A specific summation identity involving ratios of $q$-Pochhammer symbols is utilized to evaluate the infinite sums.
  • Mathematical Induction: The primary proof technique for establishing the recurrence relations and closed forms of the rational coefficients ($P_k(q)$, $V_k(q)$, $W_k(q)$, $T_k(q)$). The authors prove the base case ($k=1$) and then assume the result for $k-1$ to prove it for $k$.

3. Key Contributions and Results

The paper presents a series of theorems that provide explicit formulas for the generating functions of the defined overpartition classes.

A. General Case (Repeated Smallest Non-Overlined Part)

• Theorem 2.1: Provides the generating function for $spt_k(n)$:
  $$\sum_{n=1}^\infty spt_k(n)q^n = P_k(q)\frac{(-q^2; q)_\infty}{(q^2; q)_\infty} + (-1)^k \frac{(q; q)_{k-1}}{(-q; q)_k}$$
  Here, $P_k(q)$ is a rational function defined recursively.
• Theorem 2.2: Gives a closed formula for the rational function $P_k(q)$ as a finite sum involving $q$-Pochhammer symbols, removing the need for recursion.
• Theorem 2.3: Addresses the signed difference function $spt_k'(n)$ (even vs. odd number of parts greater than $s(\pi)$), showing its generating function simplifies to a difference of two infinite products.

B. Parity-Restricted Case

• Theorem 2.4: Provides the generating function for $spt_k^o(n)$ (where non-$s(\pi)$ parts have different parity). The result is a linear combination of three terms involving different $q$-Pochhammer products, with coefficients $V_k(q)$ and $W_k(q)$.
• Theorem 2.5: Provides closed formulas for the rational functions $V_k(q)$ and $W_k(q)$.
• Theorem 2.6 & 2.7: Address the signed difference function $spt_k'^o(n)$, providing a generating function involving a rational coefficient $T_k(q)$ and a closed formula for $T_k(q)$.

C. Applications and Corollaries

The authors demonstrate how these general theorems yield specific combinatorial identities for small values of $k$ (specifically $k=1$ and $k=2$).

• Corollary 4.1: For $k=1$, the sum $spt_1(n) + spt_1(n-1)$ equals the number of overpartitions of $n$ with no non-overlined 1s.
• Corollary 4.3: For the parity-restricted case with $k=1$, the sum $spt_1^o(n) + spt_1^o(n-2)$ relates to overpartitions into even parts and odd parts with specific restrictions.
• Corollary 4.4: Establishes a relationship for the signed difference function involving the difference between overpartitions with even and odd numbers of parts.

4. Significance

• Extension of Classical Results: The paper successfully lifts the work of Andrews and Bachraoui from standard integer partitions to overpartitions, a more complex and structurally rich combinatorial object.
• Structural Insight: By expressing these generating functions as linear combinations of $q$-Pochhammer symbols with rational coefficients, the authors reveal deep structural connections between these specific overpartition classes and classical modular forms.
• New Identities: The derived corollaries provide new identities relating specific overpartition counts to other well-known partition functions (e.g., partitions into distinct parts, partitions into even/odd parts).
• Methodological Framework: The recursive approach combined with the Andrews-Subbarao-Vidyasagar formula offers a robust template for analyzing other restricted overpartition problems.

5. Conclusion and Future Work

The authors conclude by noting an interesting pattern in the degrees of the polynomials in the numerators and denominators of the rational functions derived. They suggest that studying the degree growth of these rational functions for general $k$ and finding combinatorial proofs (as opposed to analytic generating function proofs) for the derived corollaries are promising directions for future research.

In summary, this paper significantly advances the theory of overpartitions by solving a specific restriction problem (repeated smallest non-overlined part) and providing explicit, closed-form generating functions that bridge combinatorial definitions with analytic $q$-series identities.