Displaying similar documents to “Distributive laws and Koszulness”

Hopf algebras and dendriform structures arising from parking functions

Jean-Christophe Novelli, Jean-Yves Thibon (2007)

Fundamenta Mathematicae

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We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension ( n + 1 ) n - 1 in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its structure, and show that it admits natural quotients and subalgebras whose graded components have dimensions respectively given by the Schröder numbers (plane trees), the Catalan numbers, and powers of 3. These smaller algebras are always bialgebras and belong to...

Operads for n -ary algebras – calculations and conjectures

Martin Markl, Elisabeth Remm (2011)

Archivum Mathematicum

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In [8] we studied Koszulity of a family t 𝒜 𝑠𝑠 d n of operads depending on a natural number n ∈ ℕ and on the degree d ∈ ℤ of the generating operation. While we proved that, for n ≤ 7 , the operad t 𝒜 𝑠𝑠 d n is Koszul if and only if d is even, and while it follows from [4] that t 𝒜 𝑠𝑠 d n is Koszul for d even and arbitrary n , the (non)Koszulity of t 𝒜 𝑠𝑠 d n for d odd and n ≥ 8 remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations. ...