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

We prove that the number of oscillating tableaux of length n with at most k   columns, starting at ∅ and ending at the one-column shape (1m)(1m), 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.