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 is equivalent to a standard Young tableau of shape .
The Young branching graph is the graded cover graph of the Young lattice. Its level consists of partitions of , 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 from the empty diagram to , with , has count . This differs from a saturated upward path, whose length must equal .
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Young's lattice is a combinatorial structure used in the representation theory of symmetric groups and, more broadly, in the study of symmetric functions and partition theory. It is formed by considering all partitions of a given integer and organizing them in a specific way. In particular, a Young diagram represents a partition, where a partition of a positive integer \( n \) is a way of writing \( n \) as a sum of positive integers, where the order of addends does not matter.