  We make a systematic study of matroidal mixed Eulerian numbers which are
certain intersection numbers in the matroid Chow ring generalizing the mixed
Eulerian numbers introduced by Postnikov. These numbers are shown to be
valuative and obey a log-concavity relation. We establish recursion formulas
and use them to relate matroidal mixed Eulerian numbers to the characteristic
and Tutte polynomials, reproving results of Huh-Katz and Berget-Spink-Tseng.
Generalizing Postnikov, we show that these numbers are equal to certain
weighted counts of binary trees. Lastly, we study these numbers for perfect
matroid designs, proving that they generalize the remixed Eulerian numbers of
Nadeau-Tewari.
