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Article Dans Une Revue Advances in Applied Mathematics Année : 2020

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Résumé

The m-Tamari lattice of F. Bergeron is an analogue of the classical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or -ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)n-1, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.
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hal-01289784 , version 1 (22-08-2022)

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Jean-Christophe Novelli, Jean-Yves Thibon. Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions. Advances in Applied Mathematics, 2020, 117, pp.102019.1-51. ⟨10.1016/j.aam.2020.102019⟩. ⟨hal-01289784⟩
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