 Oscillating path in the Young lattice

An oscillating tableau is a sequence of Young diagrams, each obtained from its predecessor by adding or removing one box. These are paths in the Young branching graph with both directions allowed. A path of length $\ell$ from the empty diagram to $\lambda ⊢n$, with $\ell -n=2m\ge 0$, has count $\ell !f_{\lambda }/(2^{m}n!m!)$. This differs from a saturated upward path, whose length must equal $n$.