Proof of a conjecture of Banerjee,Bringmann and Bachraoui on infinite families of congruences

1. Problem Statement

The paper addresses a specific problem in the theory of integer partitions, focusing on restricted two-color partitions.

• Context: Andrews and Bachraoui previously defined a sequence $c(k, n)$ counting partitions of $n$ into two colors (blue and red) subject to specific constraints: the smallest part is odd and occurs in blue; even blue parts are large relative to the smallest part; and even parts of the same color are distinct.
• Limiting Sequence: They studied the generating function $C_k(q)$ and its limit as $k \to \infty$, denoted as $C(q) = \sum c(n)q^n$.
• The Conjecture: Banerjee, Bringmann, and Bachraoui (BBB) recently proved several specific congruences for $c(n)$ modulo 4 and 8. However, they posed a broader conjecture (Conjecture 1.5) regarding infinite families of congruences for $c(n)$ modulo 4 and 8.
  • Specifically, they conjectured that for all integers $n, k \geq 0$:
    1. $c\left(\frac{2^{2k+3}n + 11 \cdot 4^k + 1}{3}\right) \equiv 0 \pmod 4$
    2. $c\left(\frac{2^{2k+3}n + 17 \cdot 4^k + 1}{3}\right) \equiv 0 \pmod 8$
    3. $c\left(\frac{2^{2k+4}n + 38 \cdot 4^k + 1}{3}\right) \equiv 0 \pmod 4$
• Significance: Proving this conjecture would not only resolve the BBB conjecture but also imply the earlier, simpler conjectures by Andrews and Bachraoui (which correspond to the case $k=0$).

2. Methodology

The authors employ a sophisticated combination of $q$-series identities, modular forms, and mock theta functions. The proof strategy relies heavily on the results established by Banerjee, Bringmann, and Bachraoui, augmented by a classical identity from Watson.

Key Mathematical Tools:

• Generating Functions: The authors utilize the explicit generating function for $C(q)$ derived by BBB, which relates $C(q)$ to the third-order mock theta function $\omega(q)$ and McIntosh's second-order mock theta function $B(q)$.
• Dissection Techniques: The core of the proof involves "dissecting" the generating functions. This means extracting terms where the power of $q$ satisfies specific congruences (e.g., extracting terms where the exponent is odd or even) to derive new generating functions for subsequences of $c(n)$.
• Watson's Identity: A crucial step involves using an identity proved by Watson relating $f(q^8)$, $\omega(-q)$, and $\omega(-q^4)$ to infinite products.
• Modular Arithmetic on Series: The authors perform algebraic manipulations on the $q$-series modulo 4 and 8. They utilize known congruences for the coefficients of mock theta functions (specifically $a_B(n)$) and properties of infinite products (e.g., $f_m \equiv f_{2m} \pmod 2$).

Proof Structure:

1. Lemma 2.1: The authors first establish a recursive relation (Lemma 2.1) connecting $c(2^{2k+2}n + \dots)$ to $c(4n+3)$ modulo 8. This is achieved by substituting Watson's identity into the BBB generating function and performing a series of dissections (extracting odd/even powers of $q$).
2. Lemma 2.2: They prove specific base cases for higher moduli (modulo 4 and 8) for arguments of the form $32n+15$, $32n+23$, and $64n+51$. This involves further dissection of the generating functions derived in Lemma 2.1.
3. Inductive Step: By combining the recursive relation from Lemma 2.1 with the base cases from Lemma 2.2, the authors use mathematical induction on $k$ to prove the general infinite families of congruences stated in Conjecture 1.5.

3. Key Contributions

• Resolution of Conjecture 1.5: The primary contribution is the rigorous proof of the infinite family of congruences proposed by Banerjee, Bringmann, and Bachraoui.
• Unification of Results: The paper demonstrates that the complex infinite families of congruences (for arbitrary $k$) are direct consequences of the specific cases ($k=0, 1$) and the recursive structure of the generating function.
• Extension of Previous Work: While BBB used deep connections to modular forms and mock theta functions to prove specific cases, Sun and Yao provide a more direct combinatorial-analytic proof using $q$-series identities to generalize these results to all $k$.

4. Results

The paper confirms the following congruences for the coefficients $c(n)$ of the limiting generating function $C(q)$:

1. Modulo 4:
   $$c\left(\frac{2^{2k+3}n + 11 \cdot 4^k + 1}{3}\right) \equiv 0 \pmod 4$$
2. Modulo 8:
   $$c\left(\frac{2^{2k+3}n + 17 \cdot 4^k + 1}{3}\right) \equiv 0 \pmod 8$$
3. Modulo 4 (Second Family):
   $$c\left(\frac{2^{2k+4}n + 38 \cdot 4^k + 1}{3}\right) \equiv 0 \pmod 4$$

These results imply the previously known congruences:

• $c(8n+4) \equiv 0 \pmod 4$
• $c(8n+6) \equiv 0 \pmod 8$
• $c(16n+13) \equiv 0 \pmod 4$
• $c(32n+23) \equiv 0 \pmod 8$

5. Significance and Future Directions

• Theoretical Impact: This work solidifies the understanding of the arithmetic properties of restricted two-color partitions. It highlights the power of combining mock theta functions with classical $q$-series identities to solve partition congruence problems.
• Methodological Value: The paper serves as a template for proving infinite families of congruences by establishing recursive relations between coefficients of different scales ($n$ vs $2^k n$).
• Future Work: The authors suggest two natural extensions:
  1. Extending these congruences to higher moduli (e.g., modulo 16, 32).
  2. Determining the arithmetic density of the set of integers $n$ for which $c(n) \equiv 0 \pmod{2^k}$ for fixed $k$. This would provide a probabilistic understanding of how frequently these partitions vanish modulo powers of 2.

In summary, Sun and Yao successfully settle a significant open problem in partition theory by leveraging advanced $q$-series techniques to prove a broad class of congruences that unify and extend previous findings.