Fertility fibres and coproduct coefficients in the LOT Hopf algebra
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| Format: | Preprint |
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2026
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| author | Zhu, Zhicheng Li, Jingtao Gao, Xing |
| author_facet | Zhu, Zhicheng Li, Jingtao Gao, Xing |
| contents | We study fibres of the fertility map $Φ$ from decorated rooted trees to decorated multi-index monomials. For a multi-index $\mathbf{k}$ of weight $-1$, the fibre $\mathcal F_{\mathbf{k}}=\{\,t:Φ(t)=\xx^{\mathbf{k}}\,\}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05542 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fertility fibres and coproduct coefficients in the LOT Hopf algebra Zhu, Zhicheng Li, Jingtao Gao, Xing Combinatorics 16T30, 05A15, 05C05 We study fibres of the fertility map $Φ$ from decorated rooted trees to decorated multi-index monomials. For a multi-index $\mathbf{k}$ of weight $-1$, the fibre $\mathcal F_{\mathbf{k}}=\{\,t:Φ(t)=\xx^{\mathbf{k}}\,\}$ 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. |
| title | Fertility fibres and coproduct coefficients in the LOT Hopf algebra |
| topic | Combinatorics 16T30, 05A15, 05C05 |
| url | https://arxiv.org/abs/2605.05542 |