1. Combinatorial Hopf algebras, noncommutative Hall–Littlewood functions, and permutation tableaux
- Author
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Novelli, J.-C., Thibon, J.-Y., and Williams, L.K.
- Subjects
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COMBINATORICS , *HOPF algebras , *NONCOMMUTATIVE algebras , *SYMMETRIC functions , *PERMUTATIONS , *ALGEBRAIC geometry , *COMBINATORIAL enumeration problems , *MATHEMATICAL formulas - Abstract
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. [Copyright &y& Elsevier]
- Published
- 2010
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