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If this question is dumb please excuse me.

Does this type of partition have a name and if so, what is it?

A sequence of partitions of an integer $\vec{\lambda}_1, \vec{\lambda}_2,....\vec{\lambda}_j $ such that the tuple of weights $(|\vec{\lambda}_1|,|\vec{\lambda}_2|,.... |\vec{\lambda}_j|)$ forms a partition of a fixed integer $n.$

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You might be thinking of plane partition. See my blog post… – John Jiang Mar 9 '11 at 17:12
This doesn't seem like a plane partition; for example, $\lambda_1 = (1,1,1), \lambda_2 = (2)$ would satisfy his definition, but it isn't a plane partition... – Simon Rose Mar 9 '11 at 18:16
up vote 3 down vote accepted

These are counted in OEIS A001970 where they are called "partitions of partitions" along with some other interpretations. As Simon noted, they do differ from the more-studied plane partitions.

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