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