Giraudo, Samuele Combinatorial operads from monoids. (English) Zbl 1308.05110 J. Algebr. Comb. 41, No. 2, 493-538 (2015). Summary: We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endofunctions, parking functions, packed words, permutations, planar rooted trees, trees with a fixed arity, Schröder trees, Motzkin words, integer compositions, directed animals, and segmented integer compositions. We also recover some already known (symmetric or not) operads: the magmatic operad, the associative commutative operad, the diassociative operad, and the triassociative operad. We provide presentations by generators and relations of all constructed nonsymmetric operads. Cited in 2 ReviewsCited in 13 Documents MSC: 05E15 Combinatorial aspects of groups and algebras (MSC2010) 05C05 Trees 55P48 Loop space machines and operads in algebraic topology Keywords:operad; monoid; rewriting Software:Sage-Combinat; SageMath; OEIS PDFBibTeX XMLCite \textit{S. Giraudo}, J. Algebr. Comb. 41, No. 2, 493--538 (2015; Zbl 1308.05110) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; or number of ordered partitions of [n]. a(n) = 3^n - 2^n. Number of ways to partition n labeled elements into pie slices each of at least 2 elements. Number of 3-level labeled linear rooted trees with n leaves. A simple grammar: rooted ordered set partitions. Integers whose decimal expansion start with 1, do not contain zeros and each successive digit to the right is at most one greater than the previous digit. Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed. Numbers with digital product = 2. Elements of the planar rooted trees sub-operad PRT of TN generated by 01. Elements of the sub-operad FCat(2) of TN generated by 00, 01, 02. Elements of the Schroeder trees sub-operad Schr of TN generated by 00, 01, 10. Elements of the Motzkin words sub-operad Motz of TN generated by 00 and 010. Elements of the integer compositions sub-operad Comp of TN_2 generated by 00 and 010. Elements of the segmented integer compositions sub-operad SComp of TN_3 generated by 00, 01, 02. Elements of the triassociative sub-operad Tri of TN generated by 01, 10 and 11. References: [1] Aguiar, M., Loday, J.-L.: Quadri-algebras. J. Pure Appl. Algebra 191(3), 205-221 (2004) · Zbl 1097.17002 · doi:10.1016/j.jpaa.2004.01.002 [2] Aho, A., Ullman, J.: Foundations of Computer Science. W. H. Freeman, New York (1994) [3] Berger, C., Moerdijk, I.: Axiomatic homotopy theory for operads. 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