[Young's Lattice](http://en.wikipedia.org/wiki/Young%27s_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](http://en.wikipedia.org/wiki/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.



Sublattices of Young's Lattice
It appears these lattices can be described as a kind of twisted product of the simple lattice $[n]=\{0,1,2,\ldots, n-1\}$ with the usual order. To construct the lattice of subsequences of the sequence of weakly decreasing positive integers $(k\_1,\ldots, k\_N)$, let $m(k)$ be the multiplicity of the integer $k$ in the given sequence and form the product $$[d(1)]\times \cdots \times [d(N)],$$
equipped with the order $(a\_1,\ldots, a\_N)\leq (b\_1,\ldots, b\_N)$ iff $$a\_N\leq b\_N$$ $$a\_N+a\_{N-1}\leq b\_N + b\_{N-1}$$ $$\ldots$$ $$a\_N+\cdots + a\_1\leq b\_N+\cdots + b\_1.$$


For example, for the sequence $(3,2,1,1)$, we have $d(1)=2$ and $d(2)=d(3)=1$, so that the lattice is given by $[2]\times [1]\times [1]$, with the order given above. 


This must be a standard construction in poset/lattice theory, please let me know if you've seen it somewhere. 


