Fertility fibres and coproduct coefficients in the LOT Hopf algebra

TL;DR

This paper analyzes the structure of fibres in the fertility map of decorated rooted trees within the LOT Hopf algebra, providing explicit formulas, recursive relations, and refinements for coproduct calculations.

Contribution

It introduces explicit formulas, functional equations, and recursive methods for counting and analyzing fibres and coefficients in the LOT Hopf algebra.

Findings

Derived explicit formulas for weighted counts of fibres.

Established recursive and functional equations for counts and coefficients.

Refined the admissible-cut formula for the coproduct in the LOT Hopf algebra.

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.