Young's Lattice is the lattice of inclusions of Young tableaux, or integer partitions.
In R. Stanley's Enumerative Combinatorics Vol. 1, Proposition 1.4.4., there is a generalization of integer partitions where you limit the number of times each positive number may occur in the partition (and this limit may be zero).
Apparently, this gives a sublattice of Young's lattice -- where can I find a discussion of such sublattices in the literature? They may be discussed somewhere in Stanley's books but I can't seem to find them.
The particular case which i'm interested in is when only a given finite set of positive integers may occur, each of which at most a specified number of times. This produces a finite self-dual sublattice with many nice properties... do these lattices have a name?
For example, if 2 is allowed at most once and 1 is allowed at most twice, then we have the lattice
(2,1,1) | (2,1) / \ (2) (1,1) \ / (1) | ( )
Another way of thinking of it would be as a lattice of subsequences of a finite sequence of weakly decreasing positive integers.
Update: such a lattice is a finite distributive lattice, so by the Birkhoff representation theorem, it is isomorphic to the lattice of lower sets of the poset of its join-irreducible elements. The posets in question are sub-posets of a certain infinite poset, but I am hoping for a description of this class of lattices which goes beyond simply identifying their Birkhoff representation.