Multiplicity of integer partitions in iterative process Let $(M_k)_{k\geq0}$ be a sequence of multisets. The multiset $M_0=\{[\:]\}$ has only one element, which is an empty sequence. For positive $k$, $M_k$ is a multiset of sequences of integers sorted in non-increasing order. For each sequence $s=\{s_1,s_2,\dots\,s_j\}\in M_{k-1}$, the multiset $M_k$ includes the sequences $\{s_1+1, s_2, \dots, s_j\}$, $\{s_1, s_2+1, \dots, s_j\}$, $\dots$, $\{s_1, s_2, \dots, s_j+1\}$, and $\{s_1, s_2, \dots, s_j, 1\}$ that one can obtain by adding 1 to one of the elements of $s$ or appending a new 1 at its end. If this process generates integer sequences that are not in non-increasing order, reorder the sequence.
Every element of $M_k$ is an integer partition of $k$. I am interested in the multiplicity $m$ of each integer partition in $M_k$, preferably without drawing out the entire tree for the generating process. Is there a way to express $m$ as a function of $s_1, s_2, \dots$?
It seems to me that this problem should be related to multinomial coefficients and/or some multinomial version of Pascal's triangle. But so far, I haven't been able to make that connection.
The example below shows the tree for the generating process of integer partitions and their resulting multiplicities $m$ up to $k=5$.

 A: With a few corrections in the last column (which would make the graphics messier), this looks like OEIS A036040, A080575, or A178867 which all seem to be related to Bell polynomials and vary by the order within rows (your columns).
This makes me think of Thomas Brylawski's "The lattice of integer partitions" (Disc. Math. 6 (1973) 201--219) where he developed a dominance ordering on partitions that's related to this operation.  This arose later in theoretical physics as the "sandpile model" which I've studied in a little field that I call partition dynamics.
The corrections: [4,1] can also come from [3,1] which gives it $m = 5$, and [3,1,1] can also come from [2,1,1] which gives it $m=10$.
A: This is not an answer, and too long for a comment, but I have been struck by the similarity of your drawing with this one:

that was published in this paper:
The Lattice of integer partitions and its infinite extension
In this work, we were interested in Brylawski's lattice of integer partitions already mentioned by Brian Hopkins in his answer. We encoded all partitions of any integer into an infinite tree, in which the $i$-th level are the partitions of $i$, and, more interesting, a sub-tree with same root also gives this set. In the picture above, for instance, the two dotted sets correspond to the partitions of $7$.
Our goal was not to count multiplicity, but I think some results in this paper may help.
