Generate Chain Sequence
Compute \(f(\mu)\)
Flag Type Explorer \((\lambda,a,\varepsilon)\to\mu\)
HLLL Parameterization: Every flagpole partition \(\mu\) is determined by \((\lambda,a,\varepsilon)\):
- \(\lambda\) = flag type (a partition)
- \(a\) = integer parameter controlling the initial 2-block in \(TI_2\)
- \(\varepsilon\in\{0,1\}\) = binary switch
Size formula: \(|\mu|=|\lambda|+\rho(\mu)-2\), where \(\rho=3+a+\lambda_1+\ell(\lambda)\).
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 \(\mu\) of size \(k\) where \(\rho(TI_2(\mu))=k+2-r\) and verify the SMALL condition: \(|\rho^{-1}(DV_+)|=2p(r)\).
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 \((v_1,v_2,\ldots,v_n)\) where \(v_1=0\) and \(v_{i+1}\le v_i+1\)
- Dyck Vectors: QDVs where all entries are non-negative
- Reduced Dyck Vector: Vector of minimum length in a Dyck class
- Key Formula: \(QDV_n(\lambda)=(0-\lambda_n,1-\lambda_{n-1},\ldots,n-1-\lambda_1)\)
Dyck Path Visualizer — visualize lattice paths from Dyck vectors
\(NU_1\) Operator (Successor Map)
- Definition: \(NU_1(\gamma)=\langle \ell+1,\gamma_1-1,\gamma_2-1,\ldots,\gamma_\ell-1\rangle\)
- Domain: \(\gamma_1\le \ell(\gamma)+2\)
- Properties: Preserves deficit, increases \(\operatorname{dinv}\) by 1
- Termination: \(NU_1\)-final objects where \(\gamma_1>\ell(\gamma)+2\)
\(ND_1\) Operator (Predecessor Map)
- Definition: \(ND_1(\gamma)=\langle\gamma_2+1,\gamma_3+1,\ldots,\gamma_\ell+1,1^{\gamma_1-\ell}\rangle\)
- Domain: \(\gamma_1\ge\ell(\gamma)\)
- Properties: Preserves deficit, decreases \(\operatorname{dinv}\) by 1
\(NU_2\) Operator (Extension Map)
- Rule (a): \([012^h A(-1)^{h-1}]\mapsto[0^h1A1^h]\), \(h\ge2\)
- Rule (b): \([012^k B(-1)^k]\mapsto[0^{k+1}B01^k]\), \(k\ge1\)
- Domain: \(NU_1\)-final objects
- Range: \(NU_1\)-initial objects
- Properties: Preserves deficit, increases \(\operatorname{dinv}\) by 1
Domains and Ranges
- Length: \(\ell(\gamma)\) is the number of parts of \(\gamma\), and \(\gamma_1\) is its largest part.
- \(NU_1\)-initial: \(\gamma_1<\ell(\gamma)\). Here \(ND_1\) is undefined.
- \(NU_1\)-final: \(\gamma_1>\ell(\gamma)+2\). Here \(NU_1\) is undefined.
- \(D_1\): \(\gamma_1\le\ell(\gamma)+2\), the domain of \(NU_1\).
- \(C_1\): \(\gamma_1\ge\ell(\gamma)\), the domain of \(ND_1\) and range of \(NU_1\).
- \(D_2\): Dyck classes matching one of the \(NU_2\) input patterns.
- \(C_2\): Dyck classes matching one of the \(NU_2\) output patterns.
- Combined maps: \(NU:D_1\cup D_2\to C_1\cup C_2\), and \(ND:C_1\cup C_2\to D_1\cup D_2\).
Chain Types
- \(NU_1\)-fragment: Finite chain terminated by \(NU_1\)-final object
- \(NU_1\)-tail: Infinite chain starting from \(TI(\mu)\)
- Second-order tail: Extended tail starting from \(TI_2(\mu)\)
- Global chain: Complete chain \(C_\mu\) for partition \(\mu\)