Samuele Giraudo

Samuele Giraudo
Université Gustave Eiffel · Laboratoire d'Informatique Gaspard-Monge

Ph.D; HDR

About

60
Publications
2,318
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234
Citations
Introduction
Samuele Giraudo currently works at the Laboratoire d'Informatique Gaspard-Monge, Université Paris-Est Marne-la-Vallée. Samuele does research in algebraic combinatorics. His current project is 'Operads in combinatorics.'
Additional affiliations
September 2012 - present
Université Gustave Eiffel
Position
  • Professor (Associate)
Education
December 2017 - December 2017
Université Gustave Eiffel
Field of study
  • Computer science
October 2009 - December 2011
Université Gustave Eiffel
Field of study
  • Computer science

Publications

Publications (60)
Preprint
Full-text available
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 ${\bf M} x_1 \to x_1 x_1$. We prove that the reflexive and transitive closure of this rewrite relation is a partia...
Preprint
Full-text available
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 ${\bf M} x_1 \to x_1 x_1$. We prove that the reflexive and transitive closure of this rewrite relation is a partia...
Preprint
Full-text available
A new hierarchy of operads over the linear spans of $\delta$-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 obtain operads whose partial compositions can be described in terms of intervals of the lattice of $\delta$...
Article
Full-text available
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 partitions, allowing us to connect number of standard Young tableaux and numbers of permutations. Here, we use...
Preprint
Full-text available
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 produce new ones. This dives the set of musical phrases into an algebraic framework since the set of multi-patter...
Preprint
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 dives musical phrases into an algebraic context since the set of multi-patterns has the structure of an operad...
Preprint
Full-text available
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 $\mathcal{M}$-decorated cliques and generalizing usual configurations of chords. By considering combinatorial subfamilies of $\...
Preprint
Full-text available
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operad...
Article
Full-text available
A syntax tree is a planar rooted tree where internal nodes are labeled on a graded set of generators. There is a natural notion of occurrence of contiguous pattern in such trees. We describe a way, given a set of generators G and a set of patterns P, to enumerate the trees constructed on G and avoiding P. The method is built around inclusion-exclus...
Preprint
We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and...
Preprint
Full-text available
We use operads, algebraic devices abstracting the notion of composition of combinatorial objects, to build pairs of graded graphs. For this, we first construct a pair of graded graphs where vertices are syntax trees, the elements of free nonsymmetric operads. This pair of graphs is dual for a new notion of duality called $\phi$-diagonal duality, si...
Preprint
Full-text available
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-Willwacher operad, and has many interesting suboperads. The latter is a natural extension of the pre-Lie operad in a...
Preprint
Full-text available
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 another one is a generalization of Tamari lattices. The three families of posets are related: they fit into a chain...
Preprint
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 generalization of the Kontsevich-Willwacher operad. This operad can be seen as a canonical operad on multigraphs, an...
Article
Full-text available
We study quotients of the magmatic operad that is the free nonsymmetric operad over one binary generator. In the linear setting, we show that the set of these quotients admits a lattice structure and we show an analog of the Grassmann formula for the dimensions of these operads. In the nonlinear setting, we define comb associative operads that are...
Chapter
Full-text available
Given permutations \(\pi \), \(\sigma _1\) and \(\sigma _2\), the permutation \(\pi \) (viewed as a string) is said to be a shuffle of \(\sigma _1\) and \(\sigma _2\), in symbols Open image in new window , if \(\pi \) can be formed by interleaving the letters of two strings \(p_1\) and \(p_2\) that are order-isomorphic to \(\sigma _1\) and \(\sigma...
Preprint
Full-text available
A syntax tree is a planar rooted tree where internal nodes are labeled on a graded set of generators. There is a natural notion of occurrence of contiguous pattern in such trees. We describe a way, given a set of generators and a set of patterns, to enumerate the trees avoiding this last. The method is built around inclusion-exclusion formulas form...
Preprint
Full-text available
We study quotients of the magmatic operad, that is the free nonsymmetric operad over one binary generator. In the linear setting, we show that the set of these quotients admits a lattice structure and we show an analog of the Grassmann formula for the dimensions of these operads. In the nonlinear setting, we define comb associative operads, that ar...
Preprint
Full-text available
The associative operad is the quotient of the magmatic operad by the operad congruence identifying the two binary trees of degree $2$. We introduce here a generalization of the associative operad depending on a nonnegative integer $d$, called $d$-comb associative operad, as the quotient of the magmatic operad by the operad congruence identifying th...
Preprint
Full-text available
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 problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand...
Article
Full-text available
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 problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand...
Book
Full-text available
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. A lot of operads involving comb...
Chapter
This chapter deals with vector spaces obtained from graded collections. A general framework for algebraic structures having products and coproducts is presented. Most of the algebraic structures encountered in algebraic combinatorics like associative, dendriform, pre-Lie algebras, and Hopf bialgebras fit into this framework. This chapter contains c...
Chapter
This chapter introduces nonsymmetric operads. Our presentation relies on the framework of graded collections and graded spaces introduced in the previous chapters. We consider here also set-operads, algebras over operads, free operads, presentations by generators and relations, Koszul duality and Koszulity of operads. At the end of the chapter, sev...
Chapter
This last chapter is devoted to review some applications of the theory of operads for enumerative prospects. To this aim, we present formal power series on operads, generalizing usual generating series. We also provide an overview on enrichments of operads: colored operads, cyclic operads, symmetric operads, and pros.
Chapter
This preliminary chapter contains general notions about combinatorics used in the rest of the book. We introduce the notion of collections of combinatorial objects and then the notions of posets and rewrite systems, which are seen as collections endowed with some extra structure.
Chapter
This second chapter is devoted to present general notions about treelike structures. We present more precisely the ones appearing in the algebraic and combinatorial context of nonsymmetric operads. Rewrite systems of syntax trees are exposed, as well as methods to prove their termination and their confluence.
Article
Full-text available
The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying algebraically the structures thus obtained (changes of bases, generating sets, presentations, morphisms, repres...
Article
Full-text available
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 objects, forests, dissections of polygons, and involutions. All this is a consequence of the definition of a gener...
Article
Full-text available
The vector space of all polygons with configurations of diagonals is endowed with an operad structure. This is the consequence of a functorial construction C introduced here, which takes unitary magmas M as input and produces operads. The obtained operads involve regular polygons with configurations of arcs labeled on M, called M-decorated cliques...
Article
Full-text available
We introduce a construction that takes as input a so-called stiff PRO and that outputs a Hopf algebra. Stiff PROs are particular PROs that can be described by generators and relations with some precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties a...
Article
Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform op-erad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of v...
Article
Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform operad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of vi...
Article
Full-text available
A new sort of combinatorial generating system, called bud generating system, is introduced. Bud generating systems are devices for specifying sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing to work with all these generating systems in...
Article
Full-text available
A new sort of combinatorial generating system, called bud generating system, is introduced. Bud generating systems are devices for specifying sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing to work with all these generating systems in...
Conference Paper
Full-text available
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 problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand...
Article
We generalize the construction of multitildes in the aim to provide multitilde operators for regular languages. We show that the underliying algebraic structure involves the action of some operads. An operad is an algebraic structure that mimics the composition of the functions. The involved operads are described in terms of combinatorial objects....
Article
Full-text available
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 problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand...
Research
Full-text available
We introduce a functor As from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction As provides a generalization of the associative operad because all of its generating operations are associative. This construction ha...
Article
We introduce a functor ${\sf As}$ from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction ${\sf As}$ provides a generalization of the associative operad because all of its generating operations are associative. This...
Article
Full-text available
We introduce, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter $\gamma$ of diassociative algebras of Loday, called $\gamma$-pluriassociative algebras. By Koszul duality of operads, we obtain a generalization of dendriform algebras, called $\gamma$-polydendriform algebr...
Article
Full-text available
An operad structure on certain bicolored noncrossing configurations in regular polygons is studied. Motivated by this study, a general functorial construction of enveloping operad, with input a colored operad and output an operad, is presented. The operad of noncrossing configurations is shown to be the enveloping operad of a colored operad of bubb...
Data
Full-text available
We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new operads as suboperads or quotients of the operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endofunctions, parking func-tions, packed...
Data
Full-text available
We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endofunctions, parking...
Article
Full-text available
An operad structure on certain bicoloured noncrossing configurations in regular polygons is studied. Motivated by this study, a general functorial construction of enveloping operad, with input a coloured operad and output an operad, is presented. The operad of noncrossing configurations is shown to be the enveloping operad of a coloured operad of b...
Article
We construct a new bigraded combinatorial Hopf algebra whose bases are indexed by square matrices with entries in the alphabet {0, 1, ..., k}, k >= 1, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of k-colored permutations of Novelli and Thibon, and the one of uniform block...
Article
Full-text available
We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schr\"oder trees,...
Article
We study a functorial construction from the category of monoids to the category of set-operads and we give some combinatorial examples of applications.
Article
NousétudionsNousétudions une construction fonctorielle de la catégorie des mono¨ıdesmono¨ıdes vers la catégorie des opérades ensemblistes et donnons des exemples combinatoires d'applications. Abstract — Constructing set-operads from monoids We study a functorial construction from the category of monoids to the category of set-operads and we give so...
Article
Full-text available
We give a new construction of a Hopf algebra defined first by Reading whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e., Baxter permutations, pairs of twin binary trees, etc.). Our construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-...
Article
Full-text available
This thesis comes within the scope of algebraic combinatorics and deals with the construction of several combinatorial and algebraic structures on different tree species. After defining an analogue of the plactic monoid whose equivalence classes are indexed by pairs of twin binary trees, we propose in this context an analogue of the Robinson-Schens...
Article
Full-text available
We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T, T'] where T and T' are balanced binary trees are isomorphic as posets to a hypercube. We introduce synchronous grammars that allow to generate tree-like structures and obtain fixed-point functional equations to enumerate th...
Article
Full-text available
We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylv...
Article
Full-text available
International audience We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals $[T_0, T_1]$ where $T_0$ and $T_1$ are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enu...

Projects

Projects (2)
Project
Study of the notions of operads and Pros as new tools for language theory
Project
Operads are algebraic structures formalizing the notions of operators with several inputs and relations between these (for instance, associativity). This theory comes from algebra but it seems to have many applications in computer science (for instance, in combinatorics as enumerating tools and in language theory).