Abstract
A new hierarchy of operads over the linear spans of \(\delta \)-cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We obtain operads whose partial compositions can be described in terms of intervals of the lattice of \(\delta \)-cliffs. These operads are very peculiar in the world of the combinatorial operads since, despite the relative simplicity for their construction, they are infinitely generated and they have nonquadratic and nonhomogeneous nontrivial relations. We provide a general construction for some of their quotients. We use it to endow the spaces of permutations, m-increasing trees, c-rectangular paths, and m-Dyck paths with operad structures. The operads on c-rectangular paths admit, as Koszul duals, operads generalizing the duplicial and triplicial operads.
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References
Aguiar, M., Livernet, M.: The associative operad and the weak order on the symmetric groups. J. Homotopy Relat. Struct. 2(1), 57–84 (2007)
Brouder, C., Frabetti, A.: QED Hopf algebras on planar binary trees. J. Algebra 1, 298–322 (2003)
Combe, C., Giraudo, S.: Three interacting families of Fuss-Catalan posets. Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 84B(22), (2020)
Combe, C., Giraudo, S.: Three Fuss-Catalan posets in interaction and their associative algebras. Combinatorial Theory (2022). https://doi.org/10.5070/C62156878
Denoncourt, H.: A refinement of weak order intervals into distributive lattices. Ann. Comb. 17(4), 655–670 (2013)
Dvoretzky, A., Motzkin, Th.: A problem of arrangements. Duke Math. J. 14(2), 305–313 (1947)
Giraudo, S.: Algebraic and combinatorial structures on pairs of twin binary trees. J. Algebra 360, 115–157 (2012)
Giraudo, S.: Combinatorial operads from monoids. J. Algebraic Combin. 41(2), 493–538 (2015)
Giraudo, S.: Operads from posets and Koszul duality. European J. Combin. 56C, 1–32 (2016)
Giraudo, S.: Pluriassociative algebras II: the polydendriform operad and related operads. Adv. Appl. Math. 77, 43–85 (2016)
Giraudo, S.: Nonsymmetric Operads in Combinatorics. Springer, Switzerland (2018)
Ginzburg, V., Kapranov, M.M.: Koszul duality for operads. Duke Math. J. 76(1), 203–272 (1994)
Hoffbeck, E.: A Poincaré-Birkhoff-Witt criterion for Koszul operads. Manuscripta Math. 131(1–2), 87–110 (2010)
Knuth, D.: The Art of Computer Programming. Volume 4, Fascicle 4. Generating all trees—History of combinatorial generation, p. 128. Addison Wesley Longman (2006)
Lehmer, D.H.: Teaching combinatorial tricks to a computer. Proc. Sympos. Appl. Math. 10, 179–193 (1960)
Leroux, P.: L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs. Comm. Algebra 39(8), 2661–2689 (2011)
López, N.D., Préville-Ratelle, L.-F., Ronco, M.: A simplicial complex splitting associativity. J. Pure Appl. Algebra 224(5), 106222 (2020)
Loday, J.-L., Ronco, M.O.: Order structure on the algebra of permutations and of planar binary trees. J. Algebraic Combin. 15(3), 253–270 (2002)
Loday, J.-L., Vallette, B.: Algebraic Operads, volume 346 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg (2012)
Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995)
Méndez, M.: Set Operads in Combinatorics and Computer Science. Springer, New York (2015)
Pirashvili, T.: Sets with two associative operations. Centr. Eur. J. Math. 1(2), 169–183 (2003)
Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences. https://oeis.org/
Stanley, R.P.: The Fibonacci lattice. Fibonacci Quart. 13(3), 215–232 (1975)
Zhang, Y., Gao, X., Guo, L.: Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras. J. Algebra 552, 134–170 (2020)
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This research has been partially supported by the projects CARPLO (ANR-20-CE40-0007) and ALCOHOL (ANR-19-CE40-0006) of the Agence nationale de la recherche.
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Combe, C., Giraudo, S. Cliff operads: a hierarchy of operads on words. J Algebr Comb 57, 239–269 (2023). https://doi.org/10.1007/s10801-022-01167-6
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DOI: https://doi.org/10.1007/s10801-022-01167-6