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Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux

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.
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https://hal.archives-ouvertes.fr/hal-00823073
Contributor : Jean-Yves Thibon Connect in order to contact the contributor
Submitted on : Thursday, May 16, 2013 - 9:53:18 AM
Last modification on : Friday, April 15, 2022 - 1:52:42 PM

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Jean-Christophe Novelli, Jean-Yves Thibon, Lauren K. Williams. Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux. Advances in Mathematics, Elsevier, 2010, 224 (4), pp.1311-1348. ⟨10.1016/j.aim.2010.01.006⟩. ⟨hal-00823073⟩

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