Uniform cumulative-band counting

The mean excitation level d(n−1)/(d+1)d(n-1)/(d+1) is an exact identity for uniform counting across a cumulative oscillator band.

Set N=n−1N=n-1 and give equal weight to each Cartesian oscillator state (a1,…,ad)(a_1,\ldots,a_d) with nonnegative integer entries and K=∑iai≤NK=\sum_i a_i\le N. The shell degeneracy and band size are

gd(k)=(k+d−1d−1),B=∑k=0Ngd(k)=S(n,d). g_d(k)=\binom{k+d-1}{d-1},\qquad B=\sum_{k=0}^{N}g_d(k)=S(n,d).

Thus the probability of a shell under this specified counting measure is gd(k)/Bg_d(k)/B, not 1/(N+1)1/(N+1).

Exact first and second moments

The factorial-moment identities

kgd(k)=d(k+d−1d),k(k−1)gd(k)=d(d+1)(k+d−1d+1) k g_d(k)=d\binom{k+d-1}{d},\qquad k(k-1)g_d(k)=d(d+1)\binom{k+d-1}{d+1}

and the hockey-stick sum give

E[K]=dNd+1,E[K(K−1)]=dN(N−1)d+2, \mathbb E[K]=\frac{dN}{d+1},\qquad \mathbb E[K(K-1)]=\frac{dN(N-1)}{d+2},

Var⁡(K)=dN(N+d+1)(d+1)2(d+2). \operatorname{Var}(K)= \frac{dN(N+d+1)}{(d+1)^2(d+2)}.

Equivalently, append a slack coordinate a0=N−Ka_0=N-K. The uniform set of weak compositions ∑i=0dai=N\sum_{i=0}^{d}a_i=N is symmetric in all d+1d+1 entries, so each has mean N/(d+1)N/(d+1). The slack is counting notation, not an additional physical spatial direction.

For an affine diagnostic f(K)=A+BKf(K)=A+BK, its counting average is therefore A+BdN/(d+1)A+BdN/(d+1). A quadratic diagnostic additionally uses the second moment. This gives exact finite-band benchmarks before any physical occupation weights are introduced.

Physical scope

The coordinates being counted describe dd equivalent ordinary spatial oscillator directions in an auxiliary flat model. Equal weighting of all states in the cumulative band is explicit model data. A single occupied shell, a coherent state, and a nonuniform mixture generally have different moments.

Applying these moments to a physical energy correction requires the perturbing operator and the relevant physical state. The counting mixture is a mathematical benchmark for those calculations.

The mode counting checks compare these expressions with direct rational shell sums and explicit weak-composition enumeration.