Hopf dreams and diagonal harmonics

This paper introduces a Hopf algebra structure on family of reduced pipe dreams. We show that this is free and cofree, construct surjection onto commutative permutations. The dream contains subalgebras with interesting sets generators Hilbert series related to subsequences Catalan numbers. Three other relevant include the Loday–Ronco complete binary trees, special lattice walks quarter plane, ν $\nu$ -trees -Tamari lattices. One motivates new notion chains in Tamari lattice, which are used present applications conjectures theory multivariate diagonal harmonics.