A Ballot-Word Formula for q-Zeta Numerators of Type-B Root Posets

Abstract

We determine the Chapoton q-Zeta numerator for the full positive-root poset of standard Lie type B_r, graded by root height minus one and normalized with Chapoton's fixed denominator convention. For every standard Lie rank r >= 1, we identify the root poset with a trapezoidal cell poset, adjoin a minimum, and construct an explicit EL-labelling of the resulting bounded lattice. Maximal chains are encoded by words of length 2r-2 whose reversals are ballot words, and the descents of the labelling are exactly the initial positions of left-runs. Chapoton's flag expansion therefore yields an explicit monomial formula for the q-Zeta numerator, proving coefficientwise nonnegativity in every Lie rank. The formula also gives the exact t-degree r-1, the highest t-coefficient q^((r-1)(r-2)), the total value binom(2r-2,r-1) at (q,t)=(1,1), and the specialization [t^k]H(1,t)=binom(r-1,k)^2. The result concerns only the complete normalized standard type-B branch and does not assert analogous statements for other Coxeter types or denominator conventions.