ar X iv : 0 80 4 . 09 95 v 3 [ m at h . C O ] 3 J un 2 00 9 COMBINATORIAL HOPF ALGEBRAS , NONCOMMUTATIVE HALL - LITTLEWOOD FUNCTIONS , AND PERMUTATION TABLEAUX

We introduce a new family of noncommutative analogues of the HallLittlewood symmetric functions. Our construction relies upon Tevlin’s bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2− 31.