Positive formulas for q-Zeta numerators of Ferrers-cell posets

Abstract

We give explicit positive formulas for Chapoton's $q$-Zeta numerators of the Ferrers-cell posets $F_{\mathbf b}=\{(i,c):1\leq i\leq r, i\leq c\leq b_i\}$, where $b_1\geq\cdots\geq b_r\geq r$, and for every interval of their minimum-augmented lattices. A constructive signed EL-labelling expresses the numerator as a descent enumerator over boundary-admissible path words. A finite transfer-matrix recursion recovers the full multivariate descent-set polynomial. For trapezoidal boundaries, Gaussian-binomial formulas describe every interval and every $t$-slice. At $q=1$, a Jacobi-polynomial transform gives simple negative zeros, strict fixed-offset interlacing, and an explicit arcsine push-forward limit. We also obtain algebraic fixed-offset generating functions and the growth rate $(1+\sqrt t)^2$ for $t\geq0$. The standard positive-root posets of types $A_r$, $B_r$, and $C_r$ are specializations, graded by root height minus one with Chapoton's fixed denominator. For types $B_r$ and $C_r$, this yields all-rank coefficientwise positivity, the reversed-ballot formula, the specialization $[t^k]\mathbb{H}_{P_r,\operatorname{rk}}(1,t)=\binom{r-1}{k}^2$, and sharp slice degrees with unique leading monomials. The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and Grok 4.6. OpenAI GPT-6 Astra was used for subsequent proof and citation review and manuscript revision. Further details appear in the disclosure at the end of the paper.