Plethysms and operads

Altres ajuts: acords transformatius de la UABWe introduce the T-construction, an endofunctor on the category of generalized operads, as a general mechanism by which various notions of plethystic substitution arise from more ordinary notions of substitution. In the special case of one-object unary operads, i.e. monoids, we recover the T-construction of Giraudo. We realize several kinds of plethysm as convolution products arising from the homotopy cardinality of the incidence bialgebra of the bar construction of various operads obtained from the T-construction. The bar constructions are simplicial groupoids, and in the special case of the terminal reduced operad Sym, we recover the simplicial groupoid of Cebrian (Algebraic Geom Topol 21(1):421-446, 2021), a combinatorial model for ordinary plethysm in the sense of Pólya, given in the spirit of Waldhausen S and Quillen Q constructions. In some of the cases of the T-construction, an analogous interpretation is possible

Plethysm Products, Element and Plus Constructions

Motivated by viewing categories as bimodule monoids over their isomorphism groupoids, we construct monoidal structures called plethysm products on three levels: that is for bimodules, relative bimodules and factorizable bimodules. For the bimodules, we work in the general setting of actions by categories. We give a comprehensive theory linking these levels to each other as well as to Grothendieck element constructions, indexed enrichments, decorations and algebras. Specializing to groupoid actions leads to applications including the plus construction. In this setting, the third level encompasses the known constructions of Baez-Dolan and its generalizations, as we prove. One new result is that that the plus construction can also be realized an element construction compatible with monoidal structures that we define. This allows us to prove a commutativity between element and plus constructions a special case of which was announced earlier. Specializing the results on the third level yield a criterion, when a definition of operad-like structure as a plethysm monoid -- as exemplified by operads -- is possible.Comment: 54 page

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

Multispecies virial expansions

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange–Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs