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Layerings

Layerings provide partial sums of the Hyper-Catalan series by different combinatorial levels.

  • Vertex layering: constraint sum m_k <= L
  • Edge layering: constraint sum (k-1) m_k <= L
  • Face layering: constraint 1 + sum (k-1) m_k <= 1+L

APIs

  • geodepoly.layerings.vertex_layering(t: Mapping[int,complex], Vmax: int) -> list[complex]
  • geodepoly.layerings.edge_layering(t: Mapping[int,complex], Emax: int) -> list[complex]
  • geodepoly.layerings.face_layering(t: Mapping[int,complex], Fmax: int) -> list[complex]

Example

from geodepoly import vertex_layering, edge_layering, face_layering

vals = {2: 0.1, 3: 0.02}
SV = vertex_layering(vals, Vmax=5)
SE = edge_layering(vals, Emax=5)
SF = face_layering(vals, Fmax=5)

SV[L], SE[L], SF[L] return the truncated sums at each layering level.

OEIS note

On the t2-only slice, the vertex-layered series recovers the Catalan numbers as coefficients: S(t2) = 1 + C1 t2 + C2 t2^2 + ... with C_n Catalan. Our tests confirm the first few by finite differences.