Partitions with multiplicity locally restricted

Question

I am interested in finding any result, paper, taxonomy, generating function, bijection... on what I call "Partitions with multiplicity locally restricted" that is partitions in which multiplicity of each part is restricted according to the multiplicity of the previous (smaller) ones in some recursive way.

For instance: Partitions in which number of 2's is smaller than number of 1's and number of 3's is smaller that the number of 2's and so on.

Additionaly I note that restriction on multiplicity does not necessarily involve downward multiplicity if additional upper bound on largest part size is also restricted.

Thanks in advance.

Answer 1

I do not completely understand what you are asking for but suppose $4+3+3+2+2+1+1+1+1+1 = 1\times4 +2\times3+2\times2+5\times 1$ is an acceptable partition of $19$

Then this can be seen as a partition of $19$ into triangle numbers, equivalent to $10+6+1+1+1$ in this example

The number of such partitions is listed in OEIS A007294, which gives some references and methods of calculation