  In this article we prove the $e$-positivity of $G_{\mathbf{\nu}}[X;q+1]$ when
$G_{\mathbf{\nu}}[X;q]$ is a vertical strip LLT polynomial. This property has
been conjectured by Alexandersson and Panova, and by Garsia, Haglund, Qiu and
Romero, and it implies several $e$-positivities conjectured by them and also by
Bergeron.
  We make use of a result of Carlsson and Mellit that shows that a vertical
strip LLT polynomial can be obtained by applying certain compositions of
operators of the Dyck path algebra to the constant $1$. Our proof gives in fact
an algorithm to expand these symmetric functions in the elementary basis, and
it shows, as a byproduct, that these compositions of operators are actually
multiplication operators.
