Title: Diamond-colored distributive lattices, move-minimizing games, and fundamental Weyl symmetric functions: The type $\mathsf{A}$ case,
Abstract: We present some elementary but foundational results concerning
diamond-colored modular and distributive lattices and connect these structures
to certain one-player combinatorial "move-minimizing games," in particular, a
so-called "domino game." The objective of this game is to find, if possible,
the least number of "domino moves" to get from one partition to another, where
a domino move is, with one exception, the addition or removal of a
domino-shaped pair of tiles. We solve this domino game by demonstrating the
somewhat surprising fact that the associated "game graphs" coincide with a
well-known family of diamond-colored distributive lattices which shall be
referred to as the "type $\mathsf{A}$ fundamental lattices." These lattices
arise as supporting graphs for the fundamental representations of the special
linear Lie algebras and as splitting posets for type $\mathsf{A}$ fundamental
symmetric functions, connections which are further explored in sequel papers
for types $\mathsf{A}$, $\mathsf{C}$, and $\mathsf{B}$. In this paper, this
connection affords a solution to the proposed domino game as well as new
descriptions of the type $\mathsf{A}$ fundamental lattices.