Simple description of the sign map for r greater than 2

Determine a simple description of the sign map sign_r(lambda) for partitions lambda with empty r-core when r>2, extending the known description sign_2(lambda)=(-1)^{odd(lambda)}.

Background

The paper studies character-value identities relating irreducible characters of wreath products G wr S_n to irreducible characters of symmetric groups. A key ingredient is the map sign_r from partitions of rn with empty r-core to {+1,-1}, which appears in the Roichman–Adin character identity.

The authors explain that sign_r(lambda) can be characterized through the sign of the symmetric-group character value at an element of cycle type (r,r,...,r). They note that for r=2 this sign has the simple formula sign_2(lambda)=(-1)^{odd(lambda)}, where odd(lambda) is the number of odd parts of lambda. They explicitly reiterate the unresolved question of finding an analogous simple description for r>2.