We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable.From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containing a canonical tableau we discover a new basis for the Hopf algebra of quasisymmetric functions, and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by Panova and Adin-King-Roichman.Generalizing our 0-Hecke action to one on skew standard reverse composition tableaux, we derive 0-Hecke modules whose quasisymmetric characteristics are the skew quasisymmetric Schur functions of Bessenrodt et al. This enables us to prove a restriction rule that reflects the coproduct formula for quasisymmetric Schur functions, which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for Schur functions.2010 Mathematics Subject Classification. Primary 05E05, 20C08; Secondary 05A05, 05A19, 05E10, 06A07, 16T05, 20F55. Bruhat order.The authors were supported in part by the National Sciences and Engineering Research Council of Canada. 1 2 VASU V. TEWARI AND STEPHANIE J. VAN WILLIGENBURG 5. 0-Hecke modules from SRCTs and quasisymmetric Schur functions 15 6. Source and sink tableaux, and the weak Bruhat order 16 7. The classification of tableau-cyclic and indecomposable modules 23 8. The canonical basis and enumeration of truncated shifted reverse tableaux 26 8.1. Dimensions of certain S α,Eα and truncated shifted reverse tableaux 28 9. Restriction rules and skew quasisymmetric Schur functions 31 10. Further avenues 35 References 35

We introduce a generalization of semistandard composition tableaux called permuted composition tableaux. These tableaux are intimately related to permuted basement semistandard augmented fillings studied by Haglund, Mason and Remmel. Our primary motivation for studying permuted composition tableaux is to enumerate all possible ordered pairs of permutations (σ 1 , σ 2 ) that can be obtained by standardizing the entries in two adjacent columns of an arbitrary composition tableau. We refer to such pairs as compatible pairs. To study compatible pairs in depth, we define a 0-Hecke action on permuted composition tableaux. This action naturally defines an equivalence relation on these tableaux. Certain distinguished representatives of the resulting equivalence classes in the special case of two-columned tableaux are in bijection with compatible pairs. We provide a bijection between two-columned tableaux and labeled binary trees. This bijection maps a quadruple of descent statistics for 2-columned tableaux to left and right ascent-descent statistics on labeled binary trees introduced by Gessel, and we use it to prove that the number of compatible pairs is (n + 1) n−1 . 14 4.2. Bijection between LD n and SPCT((2 n )) 15 4.3. Plane binary trees, unlabeled and labeled 162010 Mathematics Subject Classification. Primary 05E10, 20C08; Secondary 05A05, 05A19, 05C20, 05E05, 05E15.

Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial$B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots , n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.

We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants aw are expressed as a sum of normalized mixed Eulerian numbers indexed naturally by reduced words of w. The description implies that the aw are positive for all permutations w ∈ Sn of length n − 1, thereby answering a question of Harada, Horiguchi, Masuda and Park. We use the same expression to establish the invariance of aw under taking inverses and conjugation by the longest word, and subsequently establish an intriguing cyclic sum rule for the numbers.We then move toward a deeper combinatorial understanding for the aw by exploiting in addition the relation to Postnikov's divided symmetrization. Finally, we are able to give a combinatorial interpretation for aw when w is vexillary, in terms of certain tableau descents. It is based in part on a relation between the numbers aw and principal specializations of Schubert polynomials.Along the way, we prove results and raise questions of independent interest about the combinatorics of permutations, Schubert polynomials and related objects.

Abstract. The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg.