Matroidal mixed Eulerian numbers

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.