A Simple Symmetry Generating Operads
Related to Rooted Planar m-ary Trees
and Polygonal Numbers
rue Roux Soignat
69003 Lyon, France
The aim of this paper is to further explore an idea from J.-L. Loday
and extend some of his results.
We impose a natural and simple symmetry on a unit
action over the most general quadratic relation which can be written.
This leads us to two families of binary, quadratic and regular operads
whose free objects are computed, as much as possible, as well as their
duals in the sense of Ginzburg and Kapranov. Roughly speaking, free
objects found here are in relation to rooted planar m-ary trees,
triangular numbers and more generally m-tetrahedral numbers,
homogeneous polynomials on $m$ commutative indeterminates over a field
K and polygonal numbers. Involutive connected ℘-Hopf
algebras are constructed.
Full version: pdf,
(Concerned with sequences
Received July 26 2006;
revised version received April 25 2007.
Published in Journal of Integer Sequences May 4 2007.
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