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A differential poset is a locally finite graded partially ordered set with a unique minimum, finite ranks, and up and down cover operators satisfying $DU-UD=rI$ for a fixed positive integer $r$. The Young lattice has $r=1$: distinct equal-rank diagrams have equally many common upper and lower covers, and each diagram has one more upper cover than lower cover.

If $DU-UD=I$, commuting every $D$ past every $U$ gives $(D+U{)}^{\ell }={\sum }_{i+j+2m=\ell }\ell !U^i{D}^j/(2^{m}i!j!m!)$. The recurrence follows from $D{U}^i=U^{i}D+i{U}^{i-1}$. Applied to the empty diagram in the Young lattice, it counts oscillating tableaux.

The Young lattice, or Young poset, has one vertex for each partition of an integer, including the empty partition, ordered by inclusion of their Young diagrams. A diagram covers another exactly when it adds one box. The Young branching graph is its graded graph: vertices at level $i$ are the partitions of $i$, and edges join diagrams differing by one box.

A descending path from $\lambda ⊢n$ to $(1)$ determines a standard Young tableau: label the successive removed boxes $n,n-1,\dots ,2$, then label the remaining box $1$. Every removed box is a Removable node of a Young diagram, so the resulting labels increase along rows and columns. Conversely, deleting boxes in decreasing label order from a standard Young tableau gives the path. Extending by the unique edge from $(1)$ to the empty diagram yields the usual path-to-tableau correspondence.

For the operator identity, compare the coefficients of each $\mu ⊢i$ in $DU(\lambda )$ and $UD(\lambda )$. If $\mu \ne \lambda$, a common upper cover exists exactly when a common lower cover exists, and each is then unique: the two diagrams differ by exchanging one box, their union is the upper cover and their intersection the lower cover. Since unions and intersections of partition diagrams are again partition diagrams, these off-diagonal coefficients agree.

The coefficient of $\lambda$ in $DU(\lambda )$ is the number of addable nodes of a Young diagram; its coefficient in $UD(\lambda )$ counts the choices of a Removable node of a Young diagram. Along the diagram's boundary the two types of corners alternate, beginning and ending with addable corners. Hence there is exactly one more addable than removable corner. For the empty partition the counts are $1$ and $0$.

Therefore the up and down operators satisfyor $D_{i+1}{U}_i-U_{i-1}{D}_i=I_i$ at level $i$. At level zero, interpret $D_0$ and the lowering contribution as zero. This makes the Young lattice a differential poset.

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.