# Is Edmonds' Matroid partitioning algorithm optimal w.r.t lexicographical order?

We all know that, given a matroid $$(E, \mathcal{I})$$, Edmonds' Matroid partitioning algorithm will result in a tuple of $$E$$-covering, pairwise-disjoint independent sets $$(I_1, ..., I_k)$$ with optimal (smallest) $$k$$.

Assume the sizes of $$I_j$$ are decreasingly sorted: $$|I_1| \ge \; ... \; \ge |I_k|$$.

My question: Is $$(|I_1|, ..., |I_k|)$$ optimal (largest) w.r.t the lexicographical order of the lexicon $$\mathbb{Z^+} = \{ 1 < 2 < 3 < ... \}$$.

If this is NOT the case, is there any algorithm finding a tuple of $$E$$-covering, pairwise-disjoint independent sets $$(I_1, ..., I_k)$$ with the optimality of $$(|I_1|, ..., |I_k|)$$ ? Potentially it may sacrifice the optimality of $$k$$.