An extraction boundary and stable edge coefficients

Definitions and mathematical statement

The assigned text, "Problem 2.1 below. 2. Enumerative Combinatorics," is not an independent mathematical problem. The supplied source analysis identifies the first phrase as the end of a remark after Problem 1.3 and the rest as the Section 2 heading. No statement of the referenced Problem 2.1 is included, so it must not be invented from this fragment.

The research aid gives a separate Coxeter-theoretic coefficient result motivated by that cross-reference. Let $W$ be an irreducible finite Coxeter group of rank $\ell\ge1$, acting on the complexified reflection representation $V$, with basic invariant degrees $d_1,\ldots,d_\ell$ and Coxeter number $h$. For an integer $k\ge0$ define the standard $q$-Fuss--Catalan polynomial \[ C_{W,k}(q)=\prod_{i=1}^\ell \frac{1-q^{kh+d_i}}{1-q^{d_i}}, \qquad N=\deg C_{W,k}=\ell kh. \] Write \[ p_W(r)=\#\{(a_1,\ldots,a_\ell)\in{\mathbb{N}}^\ell: \sum_i d_i a_i=r\} =\dim{\mathbb{C}}[V]^W_r, \] with $p_W(r)=0$ for negative integers $r$. For $c_r=[q^r]C_{W,k}(q)$, the reported full inclusion--exclusion formula is \[ c_r=\sum_{S\subseteq\{1,\ldots,\ell\}}(-1)^{|S|} p_W\left(r-\sum_{i\in S}(kh+d_i)\right). \] The product is reciprocal, $c_{N-r}=c_r$ for $0\le r\le N$. Since the least invariant degree is $2$ and is unique for irreducible $W$, the low window is \[ c_r=p_W(r)\quad(0\le r\le kh+1), \] and the high window is \[ c_{N-r}=p_W(r)\quad(0\le r\le\min\{kh+1,N\}). \] The first low-edge defect is exactly \[ c_{kh+2}=p_W(kh+2)-1. \] Coefficients beyond polynomial degree are understood as zero, and the high-index formula is only used when its exponent is nonnegative. These identities describe edge coefficients, not equality of the polynomial with the whole invariant-ring Hilbert series. They are an auxiliary supplied result and do not recover or solve a missing independent Problem 2.1.

Short English statement

The record is a cross-reference and section heading, not a posed question. Its separate research aid gives an exact inclusion--exclusion formula for Coxeter $q$-Fuss--Catalan coefficients and identifies their initial and final Hilbert-function windows.

Sources

• American Institute of Mathematics workshop Braid groups, clusters and free probability, https://aimath.org/WWN/braidgroups/braidgroups.pdf. The supplied analysis places the fragment at the end of a remark after Problem 1.3 and before Section 2, "Enumerative Combinatorics."
• Local UnsolvedMath registry, permanent ID 20002705, source code catalogue entry, canonical location aim-probability-notes.json, notes[146], record dated 14 August 2026. The original literature field is empty.
• Supplied research attempt 1: context only, problem status at run not a problem, rigor labelled proved, novelty confidence low. The separate Coxeter coefficient result is machine-generated research aid, not peer reviewed; independently verify before citation. No separate authoritative reference for it or the absent Problem 2.1 is supplied.