On bar lengths in partitions

We present, given an odd integer d, a decomposition of the multiset of bar lengths of a bar partition λ as the union of two multisets, one consisting of the bar lengths in its d-core partition cd(λ) and the other consisting of modified bar lengths in its d-quotient partition. In particular, we obtain that the multiset of bar lengths in cd(λ) is a sub-multiset of the multiset of bar lengths in λ. Also, we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of $\mathfrak{S}_n$. The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.