On a Combinatorial Puzzle Arising from the Theory of Lascoux Polynomials

Abstract

Lascoux polynomials are a class of nonhomogeneous polynomials which form a basis of the full polynomial ring. Recently, Pan and Yu showed that Lascoux polynomials can be defined as generating polynomials for certain collections of diagrams consisting of unit cells arranged in the first quadrant generated from an associated "key diagram" by applying sequences of "$K$-Kohnert moves". Within diagrams generated in this manner, certain cells are designated as special and referred to as "ghost cells". Given a fixed Lascoux polynomial, Pan and Yu established a combinatorial algorithm in terms of "snow diagrams" for computing the maximum number of ghost cells occurring in a diagram defining a monomial of the given polynomial; having this value allows one to determine the total degree of the given Lascoux polynomial. In this paper, we study the analogous combinatorial puzzle which arises when one generalizes Lascoux polynomials to $K$-Kohnert polynomials of arbitrary diagrams. Specifically, given an arbitrary diagram, we consider the question of determining the maximum number of ghost cells contained within a diagram among those formed from our given initial one by applying sequences of $K$-Kohnert moves. In this regard, we establish means of computing the aforementioned max ghost cell value for various families of diagrams as well as for diagrams in general when one takes a greedy approach.