Deeper Floors of the Binomial Lattice

\(S(n,d) = \binom{n+d-1}{d}\) is the Ehrhart polynomial of the standard \(d\)-simplex: it counts the lattice points in the \((n-1)\)-fold dilation. The simplex's \(h^*\)-vector is \((1,0,\dots,0)\) — the cleanest possible — which is exactly why the mass formula is so rigid: any deformation of \(S(n,d)\) that drives \(h^*\) negative exits the world of lattice polytopes, and the hockey-stick identity, the Pieri rule, and the integrality of mode counts all break at once.

The trivial automorphism group of the tower has a geometric home too: the tower DAG is the Bruhat order on Schubert cells of a Grassmannian, and that order admits no non-trivial order-preserving automorphisms.

\(S(n,d)\) counts semistandard Young tableaux of shape \((n-1)\) with entries \(\le d+1\). The Robinson–Schensted–Knuth algorithm gives a deterministic combinatorial fusion rule for combining modes.

Inserting the word for one mode into another mode's tableau via RSK yields the fused mode's tableau, with the resulting shape identifying the landing sector — no Clebsch–Gordan coefficients. The \(\rho\)-meson index \(n_\rho = n_s + n_u + 2n_d = 9\) is the weight of the tableau obtained by fusing the constituent seed words: the Pieri rule for tableaux, forced by the same seed algebra that fixes \(n_s = 4\).

\(S(n,d) = \dim \mathrm{Sym}^{n-1}(\mathbb{C}^{d+1})\): the mode count at level \(n\) is the dimension of the \((n-1)\)-th symmetric power of the defining representation of \(A_d = \mathfrak{su}(d+1)\). Each mode corresponds to a weight of that representation — a degree-\((n-1)\) monomial in the \(d+1\) sector coordinates — and the Weyl group \(S_{d+1}\), permuting the coordinates, supplies the observed symmetry of the count.

The Weyl character formula gives the generating function \(\prod_i (1-x_i t)^{-1}\), which at \(x_i = 1\) reduces to the sector series \(1/(1-t)^{d+1}\) whose \(t^{n-1}\) coefficient is \(S(n,d)\).

The divided-power Hopf algebra with basis \(x_{n,d}\) has structure constants given by the hockey-stick numbers. Its antipode yields the algebraic inverse of the mass law via signed Möbius inversion.

This gives a closed-form, search-free inverse: cocommutativity makes forward and backward traversal of the hockey-stick lattice algebraically equivalent. The tower's own direction is dynamical — operand dependency in the condensation order — not algebraic.

These deeper floors show that the rigid 15-mode tower is not a numerical accident but the natural realization of structures that appear across polytopes, topology, tableaux, Lie theory, and Hopf algebras.

We now have powerful new levers: RSK for fusion rules, Hopf antipodes for exact inverses, and root-system methods for symmetries — all purely combinatorial.

See also Inevitable Structure, Further Shelves, and The Combinatorial Skeleton.