A thickening map on integer partitions

Question

I am looking for the name of the following map $t(\mu)$, defined for integer partitions $\mu=\mu_1\geq\mu_2\geq\dots$:

1. if $\mu$ is empty, return $\mu$.
2. if the first part $\mu_1$ of $\mu$ is at least the number of parts of $\mu$, return the partition $\mu_1, t(\mu_2,\dots)$
3. otherwise, let $\lambda=\mu^t$ and return $\lambda_1, t(\lambda_2,\dots)$.

One property of $t$ is that $t(\mu)$ dominates $\mu$, but I'm not sure whether this is the maps natural context.

Answer 1

The partitions produced by the map are called superdiagonal partitions in OEIS, so absent a better reference you could call it the superdiagonalising map.