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

Abstract

We establish explicit positive formulas for Chapoton's q-Zeta numerators of Ferrers-cell posets F_b = {(i,c): 1 <= i <= r, i <= c <= b_i}, for every integer r >= 1 and every boundary sequence b_1 >= ... >= b_r >= r, together with formulas for every interval of their minimum-augmented lattices. A constructive signed EL-labelling identifies the numerators with weighted descent enumerators of admissible path words, and a finite transfer-matrix recursion recovers the full multivariate descent-set polynomial. The standard positive-root posets of types A, B, and C occur as specializations, graded by root height minus one with Chapoton's fixed denominator normalization. In particular, the type-B and type-C specializations give coefficientwise positivity in every Lie rank, the reversed-ballot formula, the Narayana-square specialization at q = 1, and sharp slice degrees. For trapezoidal boundaries, Gaussian-binomial formulas describe all intervals and coefficient slices. At q = 1, the global trapezoidal profiles admit a Jacobi-polynomial description, yielding simple negative zeros, strict interlacing along fixed-offset sequences, algebraic generating functions, and an explicit limiting zero distribution. These results give a uniform combinatorial explanation of q-Zeta positivity across this family.