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 selfdual 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 joinirreducible elements. The posets in question are subposets 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.