Pipe dreams are combinatorial objects that compute Grothendieck polynomials. We introduce a new combinatorial object that naturally recasts the pipe dream formula. From this, we obtain the first direct combinatorial formula for the top degree components of Grothendieck polynomials, also known as the Castelnuovo-Mumford polynomials. We also prove the inverse fireworks case of a conjecture of Meszaros, Setiabrata, and St. Dizier on the support of Grothendieck polynomials.

The dual Cauchy identity for Schur polynomials is a fundamental result in symmetric function theory and representation theory. It states that the sum of products of two Schur polynomials indexed by conjugate partitions, in two sets of variables, equals the generating function of binary matrices. Combinatorially, this identity is realized through the dual RSK insertion, which provides a bijection between such matrices and pairs of tableaux. 

Schubert polynomials, often seen as non-symmetric generalizations of Schur polynomials, satisfy a Cauchy-type formula involving triangular binary matrices. We present an explicit insertion algorithm that establishes a bijection realizing this identity using the Pieri rule. Remarkably, our algorithm retains key features of the classical RSK and naturally involves traversals of increasing binary trees. This talk is based on ongoing joint work with Johnny Gao and Sylvester Zhang.