Chapoton, Frédéric; Hivert, Florent; Novelli, Jean-Christophe A set-operad of formal fractions and dendriform-like sub-operads. (English) Zbl 1371.18004 J. Algebra 465, 322-355 (2016). Authors’ abstract: We introduce an operad of formal fractions, abstracted from the Mould operads and containing both the Dendriform and the Tridendriform operads. We consider the smallest set-operad contained in this operad and containing four specific elements of arity two, corresponding to the generators and the associative elements of the Dendriform and Tridendriform operads. We obtain a presentation of this operad (by binary generators and quadratic relations) and an explicit combinatorial description using a new kind of bi-coloured trees. Similar results are also presented for related symmetric operads. Reviewer: Fernando Lucatelli Nunes (Coimbra) Cited in 2 Documents MSC: 18D50 Operads (MSC2010) Keywords:algebraic combinatorics; operads; dendriform Software:OEIS; operads; SageMath; Sage-Combinat PDFBibTeX XMLCite \textit{F. Chapoton} et al., J. Algebra 465, 322--355 (2016; Zbl 1371.18004) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Number of labeled rooted trees with n nodes: n^(n-1). Number of labeled rooted trees of subsets of an n-set. a(n) = binomial(3*n+1,n)/(n+1). Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers). Number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon. Number of labeled series-parallel graphs with n edges. Number of labeled rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.) Number of series-reduced planted trees with n leaves of 2 colors. 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