weighted-partitions-interval-restrictions

name	weighted-partitions-interval-restrictions
description	Exact formulas and bivariate master identity methodology for weighted partitions with interval restrictions. Use when: deriving generating functions for restricted partition functions, proving partition coefficient bounds via Rogers-Fine evaluation, establishing master identities with auxiliary variables, or analyzing quantum modular forms from interval-restricted partitions. Activation: weighted partitions, interval restrictions, bivariate master identity, Rogers-Fine evaluation, false theta partition, Andrews El Bachraoui, partition coefficient bounds, quantum modular partition
metadata	{"arxiv_id":"2606.11011","published":"2026-06-09","authors":"George E. Andrews, Mohamed El Bachraoui, Aritram Dhar, Ankush Goswami, Runqiao Li"}

Weighted Partitions with Interval Restrictions

Source: arXiv:2606.11011 — "Weighted partitions with interval restrictions: exact formulas and a bivariate master identity" by Andrews et al. (2026-06-09)

Overview

Proves two conjectures for signed partition functions a₂''(n) and b₂''(n) introduced by Andrews and El Bachraoui for interval-restricted partitions. Central result: bivariate master identity with auxiliary variable z recording number of non-compulsory parts.

Core Methodology

Partition Functions with Interval Restrictions

• a₂''(n): Signed partition function where parts > 1 are controlled by smallest even part
• b₂''(n): Companion coefficients taking values only in {-1, 0, 1, 2}

Bivariate Master Identity

Introduce auxiliary variable z tracking non-compulsory parts > 1:

(1 + q²)·B(z, q) - (1 + q)·A(z, q) = -q⁴/(1 - q³)

Where A(z, q) and B(z, q) are generating functions for a₂''(n) and b₂''(n).

Key Techniques

1. Analytic approach: Rogers-Fine evaluation at z = -1 yields false theta formula
2. Combinatorial approach: Direct coefficient description of b₂''(n)
3. Heine-Rogers-Fine proof: Independent proof of the false theta formula
4. Quantum modular interpretation: Connection to quantum modular forms

Results

• Generating function for a₂''(n): Elementary rational term + false theta series with periodic signs
• Coefficient range for b₂''(n): Proven to be exactly {-1, 0, 1, 2}
• Exact coefficient description: Explicit formula for each b₂''(n)

Implementation Steps

Step 1: Define the Partition Functions

def a2_double_prime(n):
    """Signed partition function for interval-restricted partitions."""
    # Parts > 1 controlled by smallest even part
    pass

def b2_double_prime(n):
    """Companion coefficients."""
    # Values in {-1, 0, 1, 2}
    pass

Step 2: Master Identity Verification

def master_identity(z, q, N_terms=50):
    """Verify (1+q²)B(z,q) - (1+q)A(z,q) = -q⁴/(1-q³)."""
    # Compute series expansions of A(z,q) and B(z,q)
    # Check identity holds coefficient by coefficient
    pass

Step 3: Rogers-Fine Evaluation

At z = -1, the identity reduces to the false theta formula for a₂''(n).

Pitfalls

1. Interval restriction complexity: The constraint "parts > 1 controlled by smallest even part" creates non-trivial combinatorial structure
2. False theta convergence: False theta series have restricted convergence domains — analytic continuation may be needed
3. Coefficient sign patterns: The periodic signs in the false theta formula require careful tracking

Activation

• weighted partitions, interval restrictions, bivariate master identity
• Rogers-Fine evaluation, false theta partition
• Andrews El Bachraoui partition
• partition coefficient bounds
• quantum modular partition
• 加权分拆, 区间限制分拆, Rogers-Fine求值