Generate Chain Sequence
Compute f(μ)
Definition 0.7: The map f acts on multiplicities:
- Even multiplicity aᵉ → (a−1)ᵉ (all parts decrease by 1)
- Odd multiplicity aᵉ → ((a−1)ᵉ⁻¹, a−2) (most drop by 1, one drops by 2)
Flag Type Explorer (λ, a, ε) → μ
HLLL Parameterization: Every flagpole partition μ is determined by (λ, a, ε):
- λ = flag type (a partition)
- a = integer parameter (number of 2s in TI₂)
- ε ∈ {0,1} = binary switch
Size formula: |μ| = |λ| + ρ(μ) − 2, where ρ = 3 + a + d(λ) + ℓ(λ) + ε
SMALL Condition Verification
Definition: A non-negative integer r is said to be SMALL with respect to k if:
$$ r \leq \left\lfloor \frac{k}{2} \right\rfloor - 2 $$
Find all partitions μ of size k where ρ(TI₂(μ)) = k+2-r and verify the SMALL condition: |ρ⁻¹(DV₊)| = 2·p(r)
Example: k=10 → max r = ⌊5⌋ - 2 = 3 (valid r ∈ {0,1,2,3})
Must satisfy: r ≤ ⌊k/2⌋ - 2
Mathematical Background
Quasi-Dyck Vectors
- Definition: Sequences (v₁, v₂, ..., vₙ) where v₁ = 0 and v_{i+1} ≤ v_i + 1
- Dyck Vectors: QDVs where all entries are non-negative
- Reduced Dyck Vector: Vector of minimum length in a Dyck class
- Key Formula: QDV_n(λ) = (0-λ_n, 1-λ_{n-1}, ..., n-1-λ_1)
NU₁ Operator (Successor Map)
- Definition: NU₁(γ) = ⟨ℓ+1, γ₁-1, γ₂-1, ..., γₗ-1⟩
- Domain: γ₁ ≤ ℓ(γ) + 2
- Properties: Preserves deficit, increases dinv by 1
- Termination: NU₁-final objects where γ₁ > ℓ(γ) + 2
ND₁ Operator (Predecessor Map)
- Definition: ND₁(γ) = ⟨γ₂+1, γ₃+1, ..., γₗ+1, 1^{γ₁-ℓ}⟩
- Domain: γ₁ ≥ ℓ(γ)
- Properties: Preserves deficit, decreases dinv by 1
NU₂ Operator (Extension Map)
- Rule (a): [012^h A(-1)^{h-1}] → [00^{h-1}1A1^h]
- Rule (b): [012^k B(-1)^k] → [00^k B01^k]
- Domain: NU₁-final objects
- Range: NU₁-initial objects
- Properties: Preserves deficit, increases dinv by 1
Chain Types
- NU₁-fragment: Finite chain terminated by NU₁-final object
- NU₁-tail: Infinite chain starting from TI(μ)
- Second-order tail: Extended tail starting from TI₂(μ)
- Global chain: Complete chain C_μ for partition μ