The ν-Tamari Lattice via ν-Trees, ν-Bracket Vectors, and Subword Complexes

We give a new interpretation of the $\nu$-Tamari lattice of Préville-Ratelle and Viennot in terms of a rotation lattice of $\nu$-trees. This uncovers the relation with known combinatorial objects such as north-east fillings, \mbox{tree-like} tableaux and subword complexes. We provide a simple description of the lattice property using certain bracket vectors of $\nu$-trees, and show that the Hasse diagram of the $\nu$-Tamari lattice can be obtained as the facet adjacency graph of certain subword complex. Finally, this point of view generalizes to multi $\nu$-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.

Hierarchical Cu pillar electrodes for electrochemical CO2 reduction to formic acid with low overpotential

Hierarchical Cu pillar electrodes have shown enhanced electrochemical performance for CO2 reduction due to their increased surface area and controlled lattice property.