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.
Best Answer
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.