Une opérade anticyclique sur les arbustes
Annales mathématiques Blaise Pascal, Volume 17 (2010) no. 1, p. 17-45

We define new combinatorial objects, called shrubs, such that forests of rooted trees are shrubs. We then introduce a structure of operad on shrubs. We show that this operad is contained in the Zinbiel operad, by using the inclusion of Zinbiel in the operad of moulds. We also prove that this inclusion is compatible with the richer structure of anticyclic operad that exists on Zinbiel and on moulds.

DOI : https://doi.org/10.5802/ambp.277
Classification:  18D50,  05C05,  06A07
Keywords: Operad, anticyclic operad, tree, permutation, fraction
@article{AMBP_2010__17_1_17_0,
     author = {Chapoton, Fr\'ed\'eric},
     title = {Une op\'erade anticyclique sur les arbustes},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {17},
     number = {1},
     year = {2010},
     pages = {17-45},
     doi = {10.5802/ambp.277},
     mrnumber = {2674653},
     zbl = {1202.18005},
     language = {fr},
     url = {http://www.numdam.org/item/AMBP_2010__17_1_17_0}
}
Chapoton, Frédéric. Une opérade anticyclique sur les arbustes. Annales mathématiques Blaise Pascal, Volume 17 (2010) no. 1, pp. 17-45. doi : 10.5802/ambp.277. http://www.numdam.org/item/AMBP_2010__17_1_17_0/

[1] Bergeron, F.; Labelle, G.; Leroux, P. Combinatorial species and tree-like structures, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications, Tome 67 (1998) (Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota) | MR 1629341 | Zbl 0888.05001

[2] Chapoton, F. On some anticyclic operads, Algebr. Geom. Topol., Tome 5 (2005), p. 53-69 (electronic) | Article | MR 2135545 | Zbl 1060.18004

[3] Chapoton, F. The anticyclic operad of moulds, Int. Math. Res. Not. IMRN (2007) no. 20, pp. Art. ID rnm078, 36 | MR 2363304 | Zbl 1149.18004

[4] Chapoton, Frédéric A bijection between shrubs and series-parallel posets, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (DMTCS Proceedings) (2008)

[5] Chapoton, Frédéric; Hivert, Florent; Novelli, Jean-Christophe; Thibon, Jean-Yves An operational calculus for the mould operad, Int. Math. Res. Not. IMRN (2008) no. 9, pp. Art. ID rnn018, 22 | MR 2429249 | Zbl 1146.18301

[6] Livernet, Muriel A rigidity theorem for pre-Lie algebras, J. Pure Appl. Algebra, Tome 207 (2006) no. 1, pp. 1-18 | Article | MR 2244257 | Zbl 1134.17001

[7] Loday, Jean-Louis Dialgebras, Dialgebras and related operads, Springer, Berlin (Lecture Notes in Math.) Tome 1763 (2001), pp. 7-66 | MR 1860994 | Zbl 0999.17002

[8] Loday, Jean-Louis; Ronco, Maria O. Combinatorial Hopf algebras (2008) (ArXiv :math :08100435)

[9] Stanley, Richard P. Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc., Tome 45 (1974), pp. 295-299 | Article | MR 351928 | Zbl 0297.05009

[10] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 62 (1999) (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | MR 1676282 | Zbl 0928.05001