Combinatorial Hopf algebras, noncommutative Hall–Littlewood functions, and permutation tableaux
Copyright policy )Combinatorial Hopf algebras, noncommutative Hall–Littlewood functions, and permutation tableaux
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
- Paris-Est Sup France
- French National Centre for Scientific Research France
- UNIVERSITE GUSTAVE EIFFEL France
- Institut Gaspard Monge France
- Harvard University United States
combinatorial Hopf algebras, Mathematics(all), Symmetric functions and generalizations, Connections of Hopf algebras with combinatorics, Combinatorial Hopf algebras, Permutation tableaux, 004, 510, Combinatorial aspects of groups and algebras, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], PASEP, Noncommutative symmetric functions, combinatorics, permutation tableaux, FOS: Mathematics, Mathematics - Combinatorics, noncommutative symmetric functions, Combinatorics (math.CO)
combinatorial Hopf algebras, Mathematics(all), Symmetric functions and generalizations, Connections of Hopf algebras with combinatorics, Combinatorial Hopf algebras, Permutation tableaux, 004, 510, Combinatorial aspects of groups and algebras, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], PASEP, Noncommutative symmetric functions, combinatorics, permutation tableaux, FOS: Mathematics, Mathematics - Combinatorics, noncommutative symmetric functions, Combinatorics (math.CO)
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