name or characterization for the “partition” lattice of integer partitions of some n? (Young lattice row partial order)

Question

Is there a name or characterization for the "partition" lattice of integer partitions of some n?

Young's Lattice depicts the integer partitions of numbers. Often Young diagrams are used in displaying the lattice as in this picture of Young's Lattice up to n=7. In this Hasse diagram an arrow mean adding 1 box in a connected position.

I'm not asking about this lattice's partial order, but about the partial order that can be imposed within each (equal rank) row of it.

For example, for $n=6$ the 11 row elements could be encoded as

6, 51, 42, 33, 411, 321, 222, 3111, 2211, 21111, 111111

These elements can be partially ordered. In the Hasse diagram of this order, an arrow means joining two partitions. Eg. since $1+1 = 2$ there is an arrow $411\to 42$. Likewise there is one $411\to 51$.

Here are diagrams I drew for partition lattices of integer partitions for 1 through 7 .

I'm guessing somebody has studied these lattices but I can't guess the right terminology to search for information or references or find that there is some other way of generating them.

These lattices come up in characterizing the monoid of endofunctions (a category) on an $n$ element set, $End(n)$. There is a map from each endofunction $f$ of $End(n)$ to the integer partition lattice $IP(n)$, $ip: End(n) \to IP(n)$. For function composition we have $ip(fg) = ip(gf)$ and $ip(fg) \ge ip(f)\vee ip(g)$.

(why are there no available tags for words I've bolded in this post?)

Answer 1

A starting point for a bibliographic search would be

Ziegler, Günter M. "On the poset of partitions of an integer." Journal of Combinatorial Theory, Series A 42.2 (1986): 215-222, doi:10.1016/0097-3165(86)90092-0

Answer 2

As a comment already noticed, your structure is not a lattice in the most commonly used mathematical sense.

There is however a lattice defined on the partition of an integer, introduced in 1973 by Brylawski, and extended into an infinite lattice of all integer partitions in our paper entitled The Lattice of integer partitions and its infinite extension. It has nice connections to the Young lattice.

For $n=7$, we obtain:

                     

See this MO post for more.