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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 aconjecture of Burrill, which it thereby establishes. We prove as well a"Knuth-type" extension stating a similar equi-enumeration result betweengeneralised oscillating tableaux and semistandard tableaux.

Bijections between oscillating tableaux and (semi)standard tableaux via growth diagrams