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arXiv · 2605.05542

Fertility fibres and coproduct coefficients in the LOT Hopf algebra

Zhicheng Zhu·Jingtao Li·Xing Gao
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Abstract

We study fibres of the fertility map $\Phi$ from decorated rooted trees to decorated multi-index monomials. For a multi-index $\mathbf{k}$ of weight $-1$, the fibre $\mathcal F_{\mathbf{k}}=\{\,t:\Phi(t)=\xx^{\mathbf{k}}\,\}$ 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 $\jmath$. 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 $\bar\partial$, 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.

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Zhicheng Zhu, Jingtao Li, Xing Gao. 2026-05-07. Fertility fibres and coproduct coefficients in the LOT Hopf algebra. https://arxiv.org/abs/2605.05542

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