2,192 research outputs found
Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions
The dual stable Grothendieck polynomials are a deformation of the Schur
functions, originating in the study of the K-theory of the Grassmannian. We
generalize these polynomials by introducing a countable family of additional
parameters, and we prove that this generalization still defines symmetric
functions. For this fact, we give two self-contained proofs, one of which
constructs a family of involutions on the set of reverse plane partitions
generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the
other classifies the structure of reverse plane partitions with entries 1 and
2.Comment: 29 pages. Ancillary files contain an alternative version with
different exposition (including some material on the diamond lemma, which is
implicit in the main version of the paper). Comments are welcome
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
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