Further Galleries of the Binomial Lattice

The braid arrangement consists of hyperplanes \(x_i = x_j\) in \(\mathbb{R}^{d+1}\). Intersecting with the simplex \(x_1 + \dots + x_{d+1} = n-1\), \(x_i \ge 0\) yields exactly \(S(n,d)\) regions.

The distance between two modes is the minimum number of hyperplanes crossed. This gives a natural metric on the generation graph. Decays that cross few hyperplanes become combinatorially favored, matching observed hierarchies without extra coupling constants.

Refining noncrossing partitions of \(\{1,\dots,d+1\}\) by total weight \(n-1\) recovers \(S(n,d)\). The generation tower is the lattice of noncrossing partitions under refinement and noncrossing closure.

This lattice has a known Möbius function and EL-labeling, supplying canonical shortest paths and a structural explanation for spectral independence.

Modes embed naturally as vertices or faces of the permutohedron. The hockey-stick recursion corresponds to walking along edges in the weak order. Graph distance equals inversion number, directly explaining the \(q\)-weights in deformations.

For \(d=3\), the permutohedron gives a beautiful 3D visualization of the \(d=3\) generation levels.

The Hall-Littlewood polynomial \(P_{(n-1)}(1^{d+1};t)\) at \(t=0\) recovers \(S(n,d)\). The full Macdonald \((q,t)\)-version supplies a two-parameter family of deformations that remains integral and log-concave for all positive parameters.

This is a ready-made, positivity-preserving regulator for any proposed mass adjustments.

Each mode corresponds to a soliton configuration in the box-ball system on a chain of length \(n+d-1\). Time evolution is governed by the combinatorial \(R\)-matrix at \(q=0\).

The conserved quantities are the soliton sizes. Collisions such as \(5+6=11\) become clean two-soliton scattering. Stable modes are those that return to themselves after evolution — a dynamical definition of stability.

• Use braid distance (inversion count) as a cheap proxy for transition amplitudes.
• Read shortest paths in the generation graph using the EL-labeling of the noncrossing partition lattice.
• Visualize the tower on the permutohedron for \(d=3\).
• Turn on Hall-Littlewood \(t\) to gently resolve minor collisions while preserving positivity.
• Run box-ball evolution to test dynamical stability of candidate modes.

These structures continue to shrink the physics input: the single scale \(m_{\rm scale}\) plus the geometry of the lattice is increasingly sufficient.

See also Deeper Floors of the Binomial Lattice, Inevitable Structure, and The Combinatorial Skeleton.