#####
Séminaire Lotharingien de Combinatoire, 80B.89 (2018), 12 pp.

# Karola Mészáros and Avery St. Dizier

# From Generalized Permutahedra to Grothendieck Polynomials via Flow Polytopes

**Abstract.**
We prove that for permutations 1π' where π' is
dominant, the Grothendieck polynomial
**G**_{1π'}(**x**) has saturated Newton
polytope and that the Newton polytope of each homogeneous
component of **G**_{1π'}(**x**) is a
generalized permutahedron. We connect these Grothendieck
polynomials to generalized permutahedra via a family of
dissections of flow polytopes. We naturally label each simplex
in a dissection by an integer sequence, called a left-degree
sequence, and show that the sequences arising from simplices
of a fixed dimension in our dissections of flow polytopes are
exactly the integer points of generalized permutahedra. This
connection of left-degree sequences and generalized
permutahedra together with the connection of left-degree
sequences and Grothendieck polynomials established in earlier
work of Escobar and the first author reveals a beautiful
relation between generalized permutahedra and Grothendieck
polynomials.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

The following versions are available: