Young's Lattice - Properties

Properties

• The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank.
• The poset Y is a lattice. The meet and the join of two partitions is given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice in which the meet and join operations are represented by intersections and unions, it is a distributive lattice.
• If a partition p covers k elements of Young's lattice for some k then it is covered by k + 1 elements. All partitions covered by p can be found by removing one of the "corners" of its Young diagram (boxes at the end both of their row and of their column). All partitions covering p can be found by adding one of the "dual corners" to its Young diagram (boxes outside the diagram that are the first such box both in their row and in their column). There is always a dual corner in the first row, and for each other dual corner there is a corner in the previous row, whence the stated property.
• If distinct partitions p and q both cover k elements of Y then k is 0 or 1, and p and q are covered by k elements. In plain language: two partitions can have at most one (third) partition covered by both (their respective diagrams then each have one box not belonging to the other), in which case there is also one (fourth) partition covering them both (whose diagram is the union of their diagrams).
• Saturated chains between ∅ and p are in a natural bijection with the standard Young tableaux of shape p: the diagrams in the chain add the boxes of the diagram of the standard Young tableau in the order of their numbering. More generally, saturated chains between q and p are in a natural bijection with the skew standard tableaux of skew shape p/q.
• The Möbius function of Young's lattice takes values 0, ±1. It is given by the formula