Une opérade anticyclique sur les arbustes
Annales Mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 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
Mots clés : Opérade, opérade anticyclique, arbre, permutation, fraction
@article{AMBP_2010__17_1_17_0,
     author = {Fr\'ed\'eric Chapoton},
     title = {Une op\'erade anticyclique sur les arbustes},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {17--45},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {17},
     number = {1},
     year = {2010},
     doi = {10.5802/ambp.277},
     mrnumber = {2674653},
     zbl = {1202.18005},
     language = {fr},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.277/}
}
Frédéric Chapoton. Une opérade anticyclique sur les arbustes. Annales Mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 17-45. doi : 10.5802/ambp.277. https://ambp.centre-mersenne.org/articles/10.5802/ambp.277/

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

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

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

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

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

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

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

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

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