65 research outputs found
Combinatorial operads from monoids
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 functions, packed words,
permutations, planar rooted trees, trees with a fixed arity, Schr\"oder trees,
Motzkin words, integer compositions, directed animals, and segmented integer
compositions. We also recover some already known (symmetric or not) operads:
the magmatic operad, the associative commutative operad, the diassociative
operad, and the triassociative operad. We provide presentations by generators
and relations of all constructed nonsymmetric operads.Comment: 42 pages. Complete version of the extended abstracts arXiv:1208.0920
and arXiv:1208.092
Operads from posets and Koszul duality
We introduce a functor 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 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 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
restricted to a certain family of posets with Hasse diagrams satisfying some
combinatorial properties is closed under Koszul duality.Comment: 40 page
Balanced binary trees in the Tamari lattice
We show that the set of balanced binary trees is closed by interval in the
Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 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 enumerating balanced tree intervals
Constructing combinatorial operads from monoids
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, Motzkin paths, integer compositions,
directed animals, etc. We also retrieve some known operads: the magmatic
operad, the commutative associative operad, and the diassociative operad.Comment: 12 page
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.Comment: 48 page
Enveloping operads and bicoloured noncrossing configurations
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 bubbles. Several suboperads are
also investigated, and described by generators and relations.Comment: 27 page
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 Fa\`a di Bruno algebra introduced by
Foissy.Comment: 35 page
Clones of pigmented words and realizations of special classes of monoids
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 structures. A functorial construction from the category of monoids to
the category of clones is introduced. The obtained clones involve words on
positive integers where letters are pigmented by elements of a monoid. By
considering quotients of these structures, we construct a complete hierarchy of
clones involving some families of combinatorial objects. This provides clone
realizations of some known and some new special classes of monoids as among
others the variety of left-regular bands, bounded semilattices, and regular
band monoids.Comment: 41 page
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