# Bijection between Standard Young Tableaux of height $\leq 2$ and $\lfloor n/2 \rfloor$-element subsets of $[n]$.

In OEIS sequence A001405, Mike Zabrocki claims that the number of Standard Young Tableaux of length $$\leq 2$$ is equal to $$\binom{n}{\lfloor n/2 \rfloor}$$.

I haven't been able to conjure up a bijection. Is there a standard bijection to know about?

• I think it must have to do with ballot sequences. See the two notes by Joerg Arndt on A000085. – saulspatz Mar 8 at 3:39
• There's a (non-bijective) proof here: amsacta.unibo.it/2443/1/bounded_height.pdf ; it also contains a reference to what seems to be the original representation-theoretic proof. – Marcus M Mar 8 at 15:52