Flagged Littlewood-Richardson tableaux and branching rule for classical groups

We give a new formula for the branching rule from ${\rm GL}_n$ to O}_n$ generalizing Littlewood's restriction formula. The is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish stable range. As an application, we combinatorial Lusztig $t$-weight multiplicity $K_{\mu 0}(t)$ type $B_n$ and $D_n$ highest weight $\mu$ $0$.