quantum-modular-forms-partitions

name: quantum-modular-forms-partitions description: "Number theory partition statistics methodology connecting restricted excludant statistics in parity-distinct partitions with quantum modular forms via q-series transformations." tags: ["number-theory", "partitions", "quantum-modular-forms", "q-series", "combinatorics"] related_skills: ["quantum-number-theory", "quantum-number-theory-algorithms"]

Quantum Modular Forms and Partition Statistics

Methodology for studying restricted excludant statistics in partition theory through the lens of quantum modular forms. Based on arXiv:2603.13915.

Core Concept

Studies restricted excludant statistics depending on parity in partitions where parts with same parity are distinct. Uses q-series transformations to show that generating functions of these partition statistics are related to quantum modular forms — bridging combinatorial number theory with quantum modular objects.

Methodology

Partition Statistics

1. Parity-distinct partitions: Partitions where parts of the same parity are distinct
2. Excludant statistics: Statistics based on excluded parts from the partition
3. Parity dependence: Different statistical behavior for odd vs. even exclusions
4. Generating functions: q-series encoding the partition statistics

q-Series Transformations

1. Rogers-Ramanujan type identities: Transformations relating different partition generating functions
2. Mock theta connections: Relations to Ramanujan's mock theta functions
3. Quantum modularity: Transformation properties under modular group action

Quantum Modular Forms

1. Definition: Functions with modular-like transformation properties but defined on rationals
2. Asymptotic expansions: Connection to partition asymptotics
3. Analytic continuation: Extension from discrete to continuous domain

Mathematical Framework

Partition generating function:
  P(q) = Σ_{n≥0} p(n) q^n

Excludant statistics generating function:
  E_k(q) = Σ_{n≥0} e_k(n) q^n
  
Quantum modular form relation:
  E_k(q) ~ f(τ)  where q = e^{2πiτ}
  f(γτ) = (cτ+d)^k f(τ) + r(τ)  (quantum modularity)

Applications

• Partition asymptotics: Precise asymptotic behavior of restricted partitions
• Mock modular forms: Connection to Ramanujan's mock theta functions
• Quantum invariants: Relations to quantum knot invariants and 3-manifold invariants
• Statistical mechanics: Partition functions of lattice models

Implementation Considerations

• q-series convergence: Radius of convergence and analytic continuation
• Modular transformation: Behavior under SL(2,Z) action
• Numerical evaluation: Efficient computation of partition statistics

Activation

Keywords: partition statistics, quantum modular forms, q-series, Rogers-Ramanujan, mock theta functions, number theory, combinatorics, excludant statistics, parity-distinct partitions, restricted partitions arXiv: 2603.13915 Categories: math.NT