  We observe that three context-free grammars of Dumont can be brought to a
common ground, via the idea of transformations of grammars, proposed by
Ma-Ma-Yeh. Then we develop a unified perspective to investigate several
combinatorial objects in connection with the bivariate Eulerian polynomials. We
call this approach the Dumont ansatz.
  As applications, we provide grammatical treatments, in the spirit of the
symbolic method, of relations on the Springer numbers, the Euler numbers, the
three kinds of peak polynomials, an identity of Petersen, and the two kinds of
derivative polynomials, introduced by Knuth-Buckholtz and Carlitz-Scoville, and
later by Hoffman in a broader context. We obtain a convolution formula on the
left peak polynomials, leading to the Gessel formula. In this framework, we are
led to the combinatorial interpretations of the derivative polynomials due to
Josuat-Verg\`es.
