Tags: #definition

Young's lattice

Young's lattice $\mathbf{Y}$ is the poset of all Young diagrams, ordered by inclusion.
The covering relations are created by adding one box to the previous Young diagram.
Has a unique minimal element $\mathrm{\varnothing }$, and note that this is an infinite poset.

It has sublattices $L_{m,n}$, the poset of all Young diagrams $\lambda \subseteq m\times n$ rectangle.

Properties

• It is a differential poset
• The rank numbers of $\mathbf{Y}$ are the number of partitions of a given size $r_n=p(n)$, and the q-binomial coefficients are the rank generating functions for $L_{m,n}$
• $f_{\lambda }$ is equal to the number of saturated chains in $\mathbf{Y}$ from $\mathrm{\varnothing }\to \lambda$
  • Note a saturated chain corresponds to a Young tableaux with the numbering given by the order in which we add the boxes

Examples

Finite Young lattices:

General Young lattice: