774 research outputs found
QSym over Sym has a stable basis
We prove that the subset of quasisymmetric polynomials conjectured by
Bergeron and Reutenauer to be a basis for the coinvariant space of
quasisymmetric polynomials is indeed a basis. This provides the first
constructive proof of the Garsia-Wallach result stating that quasisymmetric
polynomials form a free module over symmetric polynomials and that the
dimension of this module is n!.Comment: 12 page
Skew row-strict quasisymmetric Schur functions
Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel’s fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function
Quasisymmetric (k; l)-hook Schur functions
We introduce a quasisymmetric generalization of Berele and Regev\u27s (k,l)-hook Schur functions. These quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. The quasisymmetric hook Schur functions can be defined as the generating function for a certain set of composition tableaux on two alphabets. We will look at the combinatorics of the quasisymmetric hook Schur functions, including an analogue of the RSK algorithm and a generalized Cauchy Identity
A combinatorial interpretation of the noncommutative inverse Kostka matrix
We provide a combinatorial formula for the expansion of immaculate
noncommutative symmetric functions into complete homogeneous noncommutative
symmetric functions. To do this, we introduce generalizations of Ferrers
diagrams which we call GBPR diagrams. We define tunnel hooks, which play a role
similar to that of the special rim hooks appearing in the
E\u{g}ecio\u{g}lu-Remmel formula for the symmetric inverse Kostka matrix. We
extend this interpretation to skew shapes and fully generalize to define
immaculate functions indexed by integer sequences skewed by integer sequences.
Finally, as an application of our combinatorial formula, we extend Campbell's
results on ribbon decompositions of immaculate functions to a larger class of
shapes.Comment: 44 pages, 15 figures; corrected typos; revised arguments in Section
6, results unchange
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