Two combinatorial puzzles arising from the theory of Kohnert polynomials

Motivated by recent work of Hanser and Mayers, we study two combinatorial puzzles arising from the theory of Kohnert polynomials. Such polynomials are defined as generating polynomials for certain collections of diagrams consisting of unit cells arranged in the first quadrant generated from an initial "seed diagram" by applying what are called "Kohnert moves". Each Kohnert move affects the position of at most cell of a diagram, attempting to move the rightmost cell of a given row to the first available position below and in the same column. In this paper, we study the combinatorial puzzles defined as follows: given a diagram $D$, form a diagram that is fixed by all Kohnert moves by applying either the fewest or most possible number of Kohnert moves. For both puzzles, we find complete solutions as well as methods for combinatorially computing the associated number of Kohnert moves in terms of the initial diagram $D$.