Young lattice ($\mathbb{Y}$)

The Young lattice is a partially ordered set whose elements are integer partitions, including the empty partition. Each partition of an integer is identified with its diagram, and the order is inclusion of Young diagrams. A cover adds exactly one box. A saturated path from the empty diagram to $\lambda$ is equivalent to a standard Young tableau of shape $\lambda$.

The Young branching graph is the graded cover graph of the Young lattice. Its level $n$ consists of partitions of $n$, and its edges add one box. It is also the branching graph for complex Specht modules under the restriction branching rule for a symmetric group.

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 $n$.