  The Pieri rule is a nonnegative, multiplicity-free formula for the Schur
function expansion of the product of an arbitrary Schur function with a single
row Schur function. Key polynomials are characters of Demazure modules for the
general linear group that generalize the Schur function basis of symmetric
functions to a basis of the full polynomial ring. We prove a nonsymmetric
generalization of the Pieri rule by giving a cancellation-free,
multiplicity-free formula for the key polynomial expansion of the product of an
arbitrary key polynomial with a single part key polynomial. Our proof is
combinatorial, generalizing the Robinson--Schensted--Knuth insertion algorithm
on tableaux to an insertion algorithm on Kohnert diagrams.
