We study a polynomial sequence $C_n(x|q)$ defined as a solution of a $q$-difference equation. This sequence, evaluated at $q$-integers, interpolates Carlitz-Riordans $q$-ballot numbers. In the basis given by some kind of $q$-binomial coefficients, the coefficients are again some $q$-ballot numbers. We obtain in a combinatorial way another curious recurrence relation for these polynomials.
We introduce an operad of formal fractions, abstracted from the Mould operads and containing both the Dendriform and the Tridendriform operads. We consider the smallest set-operad contained in this operad and containing four specific elements of arity two, corresponding to the generators and the associative elements of the Dendriform and Tridendriform operads. We obtain a presentation of this operad (by binary generators and quadratic relations) and an explicit combinatorial description using a new kind of bi-colored trees. Similar results are also presented for related symmetric operads.
This paper is a brief mathematical excursion which starts from quantum electrodynamics and leads to the Moebius function of the Tamari lattice of planar binary trees, within the framework of groups of tree-expanded series. First we recall Brouders expansion of the photon and the electron Greens functions on planar binary trees, before and after the renormalization. Then we recall the structure of Connes and Kreimers Hopf algebra of renormalization in the context of planar binary trees, and of their dual group of tree-expanded series. Finally we show that the Moebius function of the Tamari posets of planar binary trees gives rise to a particular series in this group.
Using representations of quivers of type A, we define an anticyclic cooperad in the category of triangulated categories, which is a categorification of the linear dual of the Diassociative operad.
We define and study a series indexed by rooted trees and with coefficients in Q(q). We show that it is related to a family of Lie idempotents. We prove that this series is a q-deformation of a more classical series and that some of its coefficients are Carlitz q-Bernoulli numbers.
Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the $NAP$ operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.
The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture of the first author about the inverse image of non-crossing trees in the dendriform operad. Finally, we explain a connection with the formalism of noncommutative symmetric functions.
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
The structure of anticyclic operad on the Dendriform operad defines in particular a matrix of finite order acting on the vector space spanned by planar binary trees. We compute its characteristic polynomial and propose a (compatible) conjecture for the characteristic polynomial of the Coxeter transformation for the Tamari lattice, which is mostly a square root of this matrix.
