Fertility fibres and coproduct coefficients in the LOT Hopf algebra

We study fibres of the fertility map $\mathrm{\Phi }$ from decorated rooted trees to decorated multi-index monomials. For a multi-index $\mathbf{k}$ of weight $-1$, the fibre ${\mathcal{F}}_{\mathbf{k}}=\{t:\mathrm{\Phi }(t)={\text{xx}}^{\mathbf{k}}\}$ consists of all rooted trees with decoration–fertility profile $\mathbf{k}$. We consider its ordinary cardinality $F_{\mathbf{k}}$, its symmetry-weighted cardinality $W_{\mathbf{k}}$, and the coefficient mass $J_{\mathbf{k}}$ appearing in the tree expansion of the transposed embedding $\mathrm{ȷ}$. We obtain an explicit formula and a functional equation for the weighted counts, and an exact multiset recursion together with a cycle-index functional equation for the ordinary counts. We also introduce coefficient generating functions for the lowering derivation $\bar{\mathrm{\partial }}$, derive recursive and transport-array formulas for the corresponding coefficients, and use them to refine the admissible-cut formula for the coproduct in the LOT Hopf algebra.