Hmmm, I don't remember reading anything in the Furminology to describe this. It's one of Barliman's favorite sitting positions. Is it a cone? An arrow? A T? Whatever it is, it doesn't look comfy!!
http://img43.photobucket.com/albums/...llge/cone1.jpg
http://img43.photobucket.com/albums/...llge/cone2.jpg
And, just for grins, here's one of him hypnotizing his audience. You are getting sleeeeeepy. You want to give him treeeeeeats....
http://img43.photobucket.com/albums/...hypnobarli.jpg
:D