Related papers: Hypertrees and embedding of the $\mathrm{FMan}$ op…
This paper generalizes the operadic construction of the Connes-Kreimer Hopf algebra of rooted trees by Moerdijk. Examples of Hopf algebras obtained in this way include the Loday-Ronco Hopf algebra of planar binary trees and the…
The incidence category $\C_{\F}$ of a family $\F$ of colored posets closed under disjoint unions and the operation of taking convex sub-posets was introduced by the author in \cite{Sz}, where the Ringel-Hall algebra $\H_{\F}$ of $\C_{\F}$…
This text, based on the author's Bachelor's thesis, introduces the theory of Algebraic Operads, a mathematical formalism that provides a unifying framework for modern algebra. We demonstrate how the fundamental theories of associative,…
This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence…
Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the Pre-Lie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a…
We investigate the tree-to-tree functions computed by "affine $\lambda$-transducers": tree automata whose memory consists of an affine $\lambda$-term instead of a finite state. They can be seen as variations on Gallot, Lemay and Salvati's…
Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi…
If $\mathfrak{g}$ is a Frobenius Lie algebra, then the spectrum of $\mathfrak{g}$ is an algebraic invariant equal to the multiset of eigenvalues corresponding to a particular operator acting on $\mathfrak{g}$. In the case of Frobenius…
This article develops, via the perspective of (arithmetic) algebraic geometry and category theory, different aspects of geometry of information. First, we describe in the terms of Eilenberg--Moore algebras over a Giry monad, the collection…
We study the pre-Lie algebra of rooted trees $(\Cal T, \rightarrow)$ and we define a pre-Lie structure on its doubling space $(V, \leadsto)$. Also, we find the enveloping algebras of the two pre-Lie algebras denoted respectively by $(\Cal…
The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture…
This paper is part of the general project of proof mining, developed by Kohlenbach. By "proof mining" we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs.…
We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of…
With the help of Rota-Baxter operators and the Groebner-Shirshov bases, we prove that any pre-Lie algebra injectively embeds into its universal enveloping preassociative algebra.
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…
In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all…
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and…
The associative operad is a central structure in operad theory, defined on the linear span of the set of permutations. We build two analogs of the associative operad on the linear span of the set of packed words which turn out to be…
A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for…
A new hierarchy of combinatorial operads is introduced, involving families of regular polygons with configurations of arcs, called decorated cliques. This hierarchy contains, among others, operads on noncrossing configurations, Motzkin…