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An extension of Tamari lattices

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.