Samuele Giraudo
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View article: Operads on graphs: extending the pre-Lie operad and general construction
Operads on graphs: extending the pre-Lie operad and general construction Open
The overall aim of this paper is to define a structure of graph operads, thus generalizing the celebrated pre-Lie operad on rooted trees. More precisely, we define two operads on multigraphs, and exhibit a non-trivial correspondence betwee…
View article: Polynomial realizations of Hopf algebras built from nonsymmetric operads
Polynomial realizations of Hopf algebras built from nonsymmetric operads Open
The natural Hopf algebra $\mathbf{N} \cdot \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We construct polynomial realizations of $\mathbf{N} \cdot \mathcal{O}$ by using al…
View article: The music box operad: Random generation of musical phrases from patterns
The music box operad: Random generation of musical phrases from patterns Open
We introduce the notion of multi-pattern, a combinatorial abstraction of polyphonic musical phrases. The interest of this approach to encode musical phrases lies in the fact that it becomes possible to compose multi-patterns in order to pr…
View article: Operads of decorated cliques II: Noncrossing cliques
Operads of decorated cliques II: Noncrossing cliques Open
A complete study of an operad $\mathrm{NC} \mathcal{M}$ of noncrossing configurations of chords introduced in previous work of the author is performed. This operad is defined on the linear span of all noncrossing $\mathcal{M}$-cliques. The…
View article: Operad Structure of Poset Matrices
Operad Structure of Poset Matrices Open
This paper examines operad structures derived from poset matrices by formulating a set of new construction rules for poset matrices. In this direction, eleven different partial composition operations will be introduced as the basis for the…
View article: Two associative operads of packed words
Two associative operads of packed words Open
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 set-th…
View article: Polynomial realizations of natural Hopf algebras of nonsymmetric operads
Polynomial realizations of natural Hopf algebras of nonsymmetric operads Open
The natural Hopf algebra $\mathcal{N} \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We introduce new bases of these Hopf algebras deriving from free operads via new lattic…
View article: Clones of pigmented words and realizations of special classes of monoids
Clones of pigmented words and realizations of special classes of monoids Open
Clones are generalizations of operads forming powerful instruments to describe varieties of algebras wherein repeating variables are allowed in their relations. They allow us in this way to realize and study a large range of algebraic stru…
View article: The combinator ${\bf M}$ and the Mockingbird lattice
The combinator ${\bf M}$ and the Mockingbird lattice Open
We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator ${\bf M}$. This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule ${\…
View article: Mockingbird lattices
Mockingbird lattices Open
We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator ${\bf M}$. This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule ${\…
View article: Three Fuss-Catalan posets in interaction and their associative algebras
Three Fuss-Catalan posets in interaction and their associative algebras Open
We introduce $\\delta$-cliffs, a generalization of permutations and increasing\ntrees depending on a range map $\\delta$. We define a first lattice structure on\nthese objects and we establish general results about its subposets. Among the…
View article: The combinator M and the Mockingbird lattice
The combinator M and the Mockingbird lattice Open
We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator ${{\mathbf{M}}}$ . This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite …
View article: Cliff operads: a hierarchy of operads on words
Cliff operads: a hierarchy of operads on words Open
A new hierarchy of operads over the linear spans of $δ$-cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We o…
View article: The music box operad: Random generation of musical phrases from patterns
The music box operad: Random generation of musical phrases from patterns Open
We introduce the notion of multi-patterns, a combinatorial abstraction of polyphonic musical phrases. The interest of this approach in encoding musical phrases lies in the fact that it becomes possible to compose multi-patterns in order to…
View article: Operads of decorated cliques I: Construction and quotients
Operads of decorated cliques I: Construction and quotients Open
We introduce a functorial construction $\mathsf{C}$ which takes unitary magmas $\mathcal{M}$ as input and produces operads. The obtained operads involve configurations of chords labeled by elements of $\mathcal{M}$, called $\mathcal{M}$-de…
View article: Generation of musical patterns through operads
Generation of musical patterns through operads Open
We introduce the notion of multi-pattern, a combinatorial abstraction of polyphonic musical phrases. The interest of this approach lies in the fact that this offers a way to compose two multi-patterns in order to produce a longer one. This…
View article: Disorders and Permutations
Disorders and Permutations Open
The additive x-disorder of a permutation is the sum of the absolute differences of all pairs of consecutive elements. We show that the additive x-disorder of a permutation of S(n), n ≥ 2, ranges from n-1 to ⌊n²/2⌋ - 1, and we give a comple…
View article: Duality of graded graphs through operads
Duality of graded graphs through operads Open
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer…
View article: Graph insertion operads
Graph insertion operads Open
Using the combinatorial species setting, we propose two new operad structures on multigraphs and on pointed oriented multigraphs. The former can be considered as a canonical operad on multigraphs, directly generalizing the Kontsevich-Willw…
View article: Three interacting families of Fuss-Catalan posets
Three interacting families of Fuss-Catalan posets Open
Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and anot…
View article: Graph operads: general construction and natural extensions of canonical operads.
Graph operads: general construction and natural extensions of canonical operads. Open
We propose a new way of defining and studying operads on multigraphs and similar objects. For this purpose, we use the combinatorial species setting. We study in particular two operads obtained with our method. The former is a direct gener…
View article: Operads on graphs: extending the pre-Lie operad and general construction
Operads on graphs: extending the pre-Lie operad and general construction Open
The overall aim of this paper is to define a structure of graph operads, thus generalizing the celebrated pre-Lie operad on rooted trees. More precisely, we define two operads on multigraphs, and exhibit a non trivial link between them and…
View article: Quotients of the Magmatic Operad: Lattice Structures and Convergent Rewrite Systems
Quotients of the Magmatic Operad: Lattice Structures and Convergent Rewrite Systems Open
We study quotients of the magmatic operad, that is the free nonsymmetric\noperad over one binary generator. In the linear setting, we show that the set\nof these quotients admits a lattice structure and we show an analog of the\nGrassmann …
View article: Algorithmic and algebraic aspects of unshuffling permutations
Algorithmic and algebraic aspects of unshuffling permutations Open
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the pr…