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We present combinatorial rules (one theorem and two conjectures) concerningthree bases of Z[x1,x2,....]. First, we prove a "splitting" rule for the basisof key polynomials [Demazure '74], thereby establishing a new positivitytheorem about them. Second, we introduce an extension of [Kohnert '90]'s"moves" to conjecture the first combinatorial rule for a certain deformation[Lascoux '01] of the key polynomials. Third, we use the same extension toconjecture a new rule for the Grothendieck polynomials [Lascoux-Schutzenberger'82].

Combinatorial rules for three bases of polynomials