On a combinatorial puzzle arising from the theory of Lascoux polynomials

TL;DR

This paper explores a combinatorial puzzle related to Lascoux polynomials, focusing on the maximum number of ghost cells in diagrams generated from arbitrary initial diagrams using Kohnert moves, extending previous work on key diagrams.

Contribution

It generalizes the computation of maximum ghost cells from key diagrams to arbitrary diagrams, providing methods for various families and a greedy approach.

Findings

Established formulas for maximum ghost cells in specific diagram families.

Developed a greedy algorithm for estimating ghost cells in general diagrams.

Extended combinatorial understanding of Lascoux polynomial diagram structures.

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 for one to determine the total degree of the given Lascoux polynomial. In this paper, we study the combinatorial puzzle which arises when one replaces key diagrams by arbitrary diagrams in the definition of Lascoux polynomials. 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.