  The Baxter permuton is a random probability measure on the unit square which
describes the scaling limit of uniform Baxter permutations. We find an explict
formula for the expectation of the Baxter permuton, i.e.\ the density of its
intensity measure. This answers a question of Dokos and Pak (2014).
  We also prove that all pattern densities of the Baxter permuton are strictly
positive, distinguishing it from other permutons arising as scaling limits of
pattern-avoiding permutations. Our proofs rely on a recent connection between
the Baxter permuton and Liouville quantum gravity (LQG) coupled with the
Schramm-Loewner evolution (SLE). The method works equally well for a
two-parameter generalization of the Baxter permuton recently introduced by the
first author, except that the density is not as explicit. This new family of
permutons, called \emph{skew Brownian permuton}, describes the scaling limit of
a number of random constrained permutations. We finally observe that in the
LQG/SLE framework, the expected proportion of inversions in a skew Brownian
permuton equals $\frac{\pi-2\theta}{2\pi}$ where $\theta$ is the so-called
imaginary geometry angle between a certain pair of SLE curves.
