arXiv subjects

Samuele Giraudo

Publications and source records attributed to Samuele Giraudo.

At least 37 records · Page 2Linked to original sources

arXiv2021

Nonsymmetric operads in combinatorics

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 operads, together with the algebraic setting and the tools accompanying it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide connections with the theory of rewrite systems. On the other hand, a lot of operads involving combinatorial objects highlight some of their properties and allow to discover new ones. This book presents the theory of nonsymmetric operads under a combinatorial point of view. It portrays the main elements of this theory and the links it maintains with several areas of computer science and combinatorics. A lot of examples of operads appearing in combinatorics are studied and some constructions relating operads with known algebraic structures are presented. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed. This text is addressed to any computer scientist or combinatorist who looks a complete and a modern description of the theory of nonsymmetric operads. Evenly, this book is intended to an audience of algebraists who are looking for an original point of view fitting in the context of combinatorics.

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arXiv2021

Duality of graded graphs through operads

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 operads, that 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 $ϕ$-diagonal duality, similar to the ones introduced by Fomin. We also provide a general way to build pairs of graded graphs from operads, wherein underlying posets are analogous to the Young lattice. Some examples of operads leading to new pairs of graded graphs involving integer compositions, Motzkin paths, and $m$-trees are considered.

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arXiv2021

Tree series and pattern avoidance in syntax trees

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 $\mathfrak{G}$ and a set of patterns $\mathcal{P}$, to enumerate the trees constructed on $\mathfrak{G}$ and avoiding $\mathcal{P}$. The method is built around inclusion-exclusion formulas forming a system of equations on formal power series of trees, and composition operations of trees. This does not require particular conditions on the set of patterns to avoid. We connect this result to the theory of nonsymmetric operads. Syntax trees are the elements of such free structures, so that any operad can be seen as a quotient of a free operad. Moreover, in some cases, the elements of an operad can be seen as trees avoiding some patterns. Relying on this, we use operads as devices for enumeration: given a set of combinatorial objects we want enumerate, we endow it with the structure of an operad, understand it in term of trees and pattern avoidance, and use our method to count them. Several examples are provided.

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arXiv2019

Quotients of the magmatic operad: lattice structures and convergent rewrite systems

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 operads indexed by nonnegative integers generalizing the associative operad. We show that the set of comb associative operads admits a lattice structure, isomorphic to the lattice of nonnegative integers equipped with the division order. Driven by computer experimentations, we provide a finite convergent presentation for the comb associative operad in correspondence with~$3$. Finally, we study quotients of the magmatic operad by one cubic relation by expressing their Hilbert series and providing combinatorial realizations.

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arXiv2019

Colored operads, series on colored operads, and combinatorial generating systems

We introduce bud generating systems, which are used for combinatorial generation. They specify sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing us to work with all these generating systems in a unified way. The theory of bud generating systems uses colored operads. Indeed, an object is generated by a bud generating system if it satisfies a certain equation in a colored operad. To compute the generating series of the languages of bud generating systems, we introduce formal power series on colored operads and several operations on these. Series on colored operads are crucial to express the languages specified by bud generating systems and allow us to enumerate combinatorial objects with respect to some statistics. Some examples of bud generating systems are constructed; in particular to specify some sorts of balanced trees and to obtain recursive formulas enumerating these.

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arXiv2018

Generalizations of the associative operad and convergent rewrite systems

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 the left and the right comb binary trees of degree $d$. We study the case $d = 3$ and provide an orientation of its space of relations by using rewrite systems on trees and the Buchberger algorithm for operads to obtain a convergent rewrite system.

math.QA↗

arXiv2018

Algorithmic and algebraic aspects of unshuffling permutations

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, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square $(213,231)$-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on directed perfect matchings, we prove that recognizing square permutations is {\bf NP}-complete.

cs.DS↗

arXiv2017

Operads in algebraic combinatorics

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, representations), to collect combinatorial information about the underlying objects. The algebraic structures the most encountered here are magmas, posets, associative algebras, dendriform algebras, Hopf bialgebras, operads, and pros. This work explores the aforementioned research direction and provides many constructions having the particularity to build algebraic structures on combinatorial objects. We develop for instance a functor from nonsymmetric colored operads to nonsymmetric operads, from monoids to operads, from unitary magmas to nonsymmetric operads, from finite posets to nonsymmetric operads, from stiff pros to Hopf bialgebras, and from precompositions to nonsymmetric operads. These constructions bring alternative ways to describe already known structures and provide new ones, as for instance, some of the deformations of the noncommutative Faà di Bruno Hopf bialgebra of Foissy and a generalization of the dendriform operad of Loday. We also use algebraic structures to obtain enumerative results. In particular, nonsymmetric colored operads are promising devices to define formal series generalizing the usual ones. These series come with several products (for instance a pre-Lie product, an associative product, and their Kleene stars) enriching the usual ones on classical power series. This provides a framework and a toolbox to strike combinatorial questions in an original way. The first two chapters pose the elementary notions of combinatorics and algebraic combinatorics used here. The last ten chapters contain our original research.

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arXiv2017

Combalgebraic structures on decorated cliques

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 general functorial construction from unitary magmas to operads. We study some of its main properties and show that this construction includes the operad of bicolored noncrossing configurations and the operads of simple and double multi-tildes. We focus in more details on a suboperad of noncrossing decorated cliques by computing its presentation, its Koszul dual, and showing that it is a Koszul operad.

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arXiv2017

Operads of decorated cliques

The vector space of all polygons with configurations of diagonals is endowed with an operad structure. This is the consequence of a functorial construction $\mathsf{C}$ introduced here, which takes unitary magmas $\mathcal{M}$ as input and produces operads. The obtained operads involve regular polygons with configurations of arcs labeled on $\mathcal{M}$, called $\mathcal{M}$-decorated cliques and generalizing usual polygons with configurations of diagonals. We provide here a complete study of the operads $\mathsf{C}\mathcal{M}$. By considering combinatorial subfamilies of $\mathcal{M}$-decorated cliques defined, for instance, by limiting the maximal number of crossing diagonals or the maximal degree of the vertices, we obtain suboperads and quotients of $\mathsf{C}\mathcal{M}$. This leads to a new hierarchy of operads containing, among others, operads on noncrossing configurations, Motzkin configurations, forests, dissections of polygons, and involutions. We show that the suboperad of noncrossing configurations is Koszul and exhibit its presentation by generators and relations. Besides, the construction $\mathsf{C}$ leads to alternative definitions of several operads, like the operad of bicolored noncrossing configurations and the operads of simple and double multi-tildes.

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arXiv2016

Combinatorial Hopf algebras from PROs

We introduce a general 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 precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties and review some examples of application. We get in particular Hopf algebras on heaps of pieces and retrieve some deformed versions of the noncommutative Faà di Bruno algebra introduced by Foissy.

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arXiv2016

Operads from posets and Koszul duality

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 construction has a very singular property: the operads obtained from ${\sf As}$ are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization, and dimensions, depend on combinatorial properties of the starting posets. Among others, we show that the property of being a forest for the Hasse diagram of the starting poset implies that the obtained operad is Koszul. Moreover, we show that the construction ${\sf As}$ restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality.

math.CO↗

arXiv2016

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a categoy of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter $γ$ of diassociative algebras, called $γ$-pluriassociative algebras, so that $1$-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with $2γ$ associative binary operations satisfying some relations. We provide a complete study of the $γ$-pluriassociative operads, the underlying operads of the category of $γ$-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in $γ$-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.

math.CO↗

arXiv2016

Pluriassociative algebras II: The polydendriform operad and related operads

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 view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter $γ$ of dendriform algebras, called $γ$-polydendriform algebras, so that $1$-polydendriform algebras are dendriform algebras. For that, we consider the operads obtained as the Koszul duals of the $γ$-pluriassociative operads introduced by the author in a previous work. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, $γ$-polydendriform algebras seem adapted structures to split associative operations into $2γ$ operation so that some partial sums of these operations are associative. We provide a complete study of the $γ$-polydendriform operads, the underlying operads of the category of $γ$-polydendriform algebras. We exhibit several presentations by generators and relations, compute their Hilbert series, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer parameter of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.

math.CO↗

arXiv2016

Pluriassociative and polydendriform algebras

We introduce, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter $γ$ of diassociative algebras of Loday, called $γ$-pluriassociative algebras. By Koszul duality of operads, we obtain a generalization of dendriform algebras, called $γ$-polydendriform algebras. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, $γ$-polydendriform algebras seem adapted structures to split associative operations into $2 γ$ operations so that some partial sums of these operations are associative. We provide a complete study of the operads governing our generalizations of the diassociative and dendriform operads. Among other, we exhibit several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.

math.CO↗

arXiv2016

Unshuffling Permutations

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, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square $(213,231)$-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on oriented perfect matchings, we prove that recognizing square permutations is $\mathbf{NP}$-complete.

cs.DS↗

arXiv2016

Operads, quasiorders, and regular languages

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. These operads are obtained from more primitive objects, namely precompositions, whose algebraic counter-parts are investigated. One of these operads acts faithfully on languages in the sense that two different operators act in two different ways.

cs.FL↗

arXiv2015

Combinatorial Hopf algebra structure on packed square matrices

We construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet $\{0, 1, ..., k\}$, $k \geq 1$, 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 permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra, we obtain a new one indexed by alternating sign matrices. We study some of its properties and algebraic quotients defined through alternating sign matrices statistics.

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