Journal of Integer Sequences, Vol. 10 (2007), Article 07.4.7

A Simple Symmetry Generating Operads Related to Rooted Planar m-ary Trees and Polygonal Numbers

Philippe Leroux
L.P. 27
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.

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(Concerned with sequences A000217 A000292 A000326 A001764 A002293 A002294 A113206 and A113207 .)

Received July 26 2006; revised version received April 25 2007. Published in Journal of Integer Sequences May 4 2007.

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