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We compute the expansion of the cohomology class of the permutahedral varietyin the basis of Schubert classes. The resulting structure constants $a_w$ areexpressed as a sum of \emph{normalized} mixed Eulerian numbers indexednaturally by reduced words of $w$. The description implies that the $a_w$ arepositive for all permutations $w\in S_n$ of length $n-1$, thereby answering aquestion of Harada, Horiguchi, Masuda and Park. We use the same expression toestablish the invariance of $a_w$ under taking inverses and conjugation by thelongest word, and subsequently establish an intriguing cyclic sum rule for thenumbers. We then move toward a deeper combinatorial understanding for the $a_w$by exploiting in addition the relation to Postnikov's divided symmetrization.Finally, we are able to give a combinatorial interpretation for $a_w$ when $w$is vexillary, in terms of certain tableau descents. It is based in part on arelation between the numbers $a_w$ and principal specializations of Schubertpolynomials. Along the way, we prove results and raise questions of independentinterest about the combinatorics of permutations, Schubert polynomials andrelated objects. We also sketch how to extend our approach to other Lie types,highlighting in particular an identity of Klyachko.

The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials