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 gi...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 224; no. 4; pp. 1311 - 1348
Main Authors Novelli, J.-C., Thibon, J.-Y., Williams, L.K.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.07.2010
Elsevier
Subjects
Combinatorial Hopf algebras
Combinatorics
Mathematics
Noncommutative symmetric functions
PASEP
Permutation tableaux
Permutation tableaux
Noncommutative symmetric functions
PASEP
Combinatorial Hopf algebras
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Summary: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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2010.01.006