17,292 research outputs found
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.Comment: 37 pages, 4 figures, new references adde
The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables
We identify a collection of primitive elements generating the Hopf algebra
NCSym of symmetric functions in noncommuting variables and give a combinatorial
formula for the antipode.Comment: 8 pages; footnote added; references added; further remarks adde
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