Abstract. We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.

We construct indecomposable modules for the 0-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions.

We introduce a shifted analog of the plactic monoid of Lascoux and Schützen-berger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

We exhibit a canonical connection between maximal (0, 1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between k-triangulations of the n-gon and k-fans of Dyck paths of length 2(n − 2k). Using this, we translate a conjectured cyclic sieving phenomenon for k-triangulations with rotation to the language of k-flagged tableaux with promotion.

We provide a direct geometric bijection for the number of lattice paths that never go below the line y ¼ kx for a positive integer k: This solution to the Generalized Ballot Problem is in the spirit of the reflection principle for the Ballot Problem (the case k ¼ 1), but it uses rotation instead of reflection. It also gives bijective proofs of the refinements of the Generalized Ballot Problem which consider a fixed number of right-up or up-right corners. r