Novelli, Jean-ChristopheThibon, Jean-YvesWilliams, Lauren2012-03-052010Novelli, Jean-Christophe, Thibon, Jean-Yves, and Lauren Williams. 2010. Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux. Advances in Mathematics 224(4): 1311–1348.0001-87081090-2082http://nrs.harvard.edu/urn-3:HUL.InstRepos:8316109We introduce a new family of noncommutative analogues of the Hall-Littlewood 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.en-UScombinatoricsJournal Article2012-03-0510.1016/j.aim.2010.01.006