I would like the tabbar to start with the first tab not with "tabs".
"tabs>1 main.tex>2 section_01.tex> <tab 2/2"
The actual behavior occurs because of the vim-airline plugin and can be removed adding the following property to your vimrc:
let g:airline#extensions#tabline#tabs_label = ''
So there will be no more leading "tabs>" in the tabline.