  Let $W$ be a finite Coxeter group and $L$ be a weight function on $W$ in the
sense of Lusztig. We have recently introduced a pre-order relation $\preceq_L$
on the set of irreducible characters of $W$ which extends Lusztig's definition
of "families" and which, conjecturally, corresponds to the ordering given by
Kazhdan--Lusztig cells. Here, we give an explicit description of $\preceq_L$
for $W$ of type $B_n$ and any $L$. (All other cases are known from previous
work.) This crucially relies on some new combinatorial constructions around
Lusztig's "symbols". Combined with previous work, we deduce general
compatibility results between $\preceq_L$ and Lusztig's $\ba$-function, valid
for any $W,L$.
