Suppose we want to partition the elements of a matroid $M$ into $n$ subsets, each of which is a base. Of course this is not always possible. For example, when $M$ is the uniform matroid on $8$ elements with capacity $6$ and $n=2$, in any partition at least one subset has fewer than 6 elements, so it is not a basis.
However, any partition of $M$ into $n$ independent sets can be extended to a partition into $n$ bases as follows:
- Add some new elements. (For $n=2$, four new elements are needed).
- Add some new independent sets, such that the previous bases remain bases. (For $n=2$, we can just add all subsets of $6$ elements from the new set of $12$ elements).
- Add the new elements to the sets of the original partition such that they become bases.
Denote the operation satisfying the properties 1, 2, 3 above as a feasible extension of the given partition.
My question is: what other matroids have this property, that each partition into $n$ independent sets has a feasible extension to a partition into $n$ bases?
One obvious example is the partition matroid. Each partition into independent sets can be extended by adding new elements to the various blocks (categories) of the partition, and distributing them among subsets with unfilled capacity. So far I could not find more examples. Are there other examples?