  Brundan and Kleshchev introduced graded decomposition numbers for
representations of cyclotomic Hecke algebras of type $A$, which include group
algebras of symmetric groups. Graded decomposition numbers are certain Laurent
polynomials, whose values at 1 are the usual decomposition numbers. We show
that in quantum characteristic 2 every such polynomial has non-zero
coefficients either only in odd or only in even degrees. As a consequence, we
find the first examples of graded decomposition numbers of symmetric groups
with non-zero coefficients in some negative degrees.
