Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux
arXiv:0804.0995
Abstract
We 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.
37 pages, 4 figures, new references added
References in corpus (6)
- Total positivity, Grassmannians, and networks
- Free quasi-symmetric functions of arbitrary level
- Construction of dendriform trialgebras
- q and q,t-Analogs of Non-commutative Symmetric Functions
- Noncommutative symmetric functions and quasi-symmetric functions with two and more paramters
- Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers