The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-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.

中文翻译：

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m-置换、(m + 1)-ary 树和 m-parking 函数的 Hopf 代数

F. Bergeron 的 m-Tamari 格是经典 Tamari 阶的类似物，该阶定义在由 Fuss-Catalan 数计数的对象上，例如 m-Dyck 路径或 (m+1)-ary 树。另一方面，Tamari 阶与平面二叉树的 Loday-Ronco Hopf 代数中的乘积有关。我们引入了基于 (m+1)-ary 树的新组合 Hopf 代数，其结构由 m-Tamari 格子描述。与平面二叉树可以解释为排列的西尔维斯特类一样，我们获得 (m+1)-ary 树作为我们称为 m 排列的西尔维斯特类。这些对象不再与递减的 (m+1)-ary 树成双射，并且称为 metasylvester 的更精细的同余允许我们基于这些递减树构建 Hopf 代数。相反，一个更粗糙的同余，称为hyposylvester，导致分级维数 (m+1)^{n-1} 的 Hopf 代数，以自然的方式推广非交换对称函数和准对称函数。最后，压缩词和停放函数的代数也接受这样的 m-类似物，我们展示了它们的子代数和由各种同余引起的商。