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Séminaire Lotharingien de Combinatoire, B74a (2015), 11 pp.

# Colleen Ross and Alexander Yong

# Combinatorial Rules for Three Bases of Polynomials

**Abstract.**
We present combinatorial rules (one theorem and two conjectures)
concerning three bases of
**Z**[*x*_{1},*x*_{2},...].
First, we prove a "splitting" rule for the basis of Key polynomials
[Demazure, *Bull. Sci. Math.* **98** (1974), 163-172],
thereby establishing a new positivity theorem about them.
Second, we introduce an extension of Kohnert's
[*Bayreuth. Math. Schriften* **38** (1990), 1-97]
"moves" to conjecture
the first combinatorial rule for a certain deformation
[Lascoux, in: *Physics and Combinatorics*,
World Scientific Publishing, 2001, pp. 164-179]
of
the Key polynomials. Third, we use the same extension to
conjecture a new rule for the Grothendieck polynomials
[Lascoux and Schützenberger, *C. R. Acad. Sci. Paris
Sér. I Math.*
**295** (1982), 629-633].

Received: November 6, 2013.
Revised: April 10, 2015.
Accepted: July 21, 2015.

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