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arXiv · 0804.0995

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

Jean-Christophe Novelli·Jean-Yves Thibon·Lauren K. Williams
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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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Jean-Christophe Novelli, Jean-Yves Thibon, Lauren K. Williams. 2009-06-03. Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux. https://arxiv.org/abs/0804.0995

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