On Ranked and Bounded Kohnert Posets

Abstract In this paper, we explore combinatorial properties of the posets associated with Kohnert polynomials. In particular, we determine a sufficient condition guaranteeing when such “Kohnert posets” are bounded and two necessary conditions for when they are ranked. Moreover, we apply the aforementioned conditions to find complete characterizations of when the Kohnert posets associated with Demazure characters are bounded and when they are ranked. In the case of ranked Kohnert posets associated with Demazure characters, our characterization is completely in terms of the underlying weak composition and motivates the definition of what we call “pure compositions”.