\BOOKMARK [1][-]{section.1}{Introduction}{}% 1
\BOOKMARK [1][-]{section.2}{Main results}{}% 2
\BOOKMARK [1][-]{section.3}{Proofs of the main results}{}% 3
\BOOKMARK [2][-]{subsection.3.1}{Eulerian numbers}{section.3}% 4
\BOOKMARK [2][-]{subsection.3.2}{Combinatorial proofs of the theorems}{section.3}% 5
\BOOKMARK [1][-]{section.4}{Acknowledgments}{}% 6