On the Support of Grothendieck Polynomials

Grothendieck polynomials \(\mathfrak {G}_w\) ${\mathfrak{G}}_w$ of permutations \(w\in S_n\) $w\in S_n$ were introduced by Lascoux and Schützenberger (C R Acad Sci Paris Sér I Math 295(11):629–633, 1982) as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the flag variety of \(\mathbb {C}^n\) ${\mathbb{C}}^n$ . We conjecture that the exponents of nonzero terms of the Grothendieck polynomial \(\mathfrak {G}_w\) ${\mathfrak{G}}_w$ form a poset under componentwise comparison that is isomorphic to an induced subposet of \(\mathbb {Z}^n\) ${\mathbb{Z}}^n$ . When \(w\in S_n\) $w\in S_n$ avoids a certain set of patterns, we conjecturally connect the coefficients of \(\mathfrak {G}_w\) ${\mathfrak{G}}_w$ with the Möbius function values of the aforementioned poset with \(\hat{0}\) $\widehat{0}$ appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations