  We prove that the number of oscillating tableaux of length $n$ with at most
$k$ columns, starting at $\emptyset$ and ending at the one-column shape
$(1^m)$, is equal to the number of standard Young tableaux of size~$n$ with $m$
columns of odd length, all columns of length at most $2k$. This refines a
conjecture of Burrill, which it thereby establishes. We prove as well a
"Knuth-type" extension stating a similar equi-enumeration result between
generalised oscillating tableaux and semistandard tableaux.
