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Abstract Given a subgroup G of the symmetric group S n , the cycle index polynomial cyc G is the average of the power-sum symmetric polynomials indexed by the cycle types of permutations in G. By Pólya's Theorem, the monomial expansion of cyc G is the generating function for weighted colorings of n objects, where we identify colorings related by one of the symmetries in G. This paper develops combinatorial formulas for the fundamental quasisymmetric expansions and Schur expansions of certain cycle index polynomials. We give explicit bijective proofs based on standardization algorithms applied to equivalence classes of colorings. Subgroups studied here include Young subgroups of S n , the alternating groups A n , direct products, conjugate subgroups, and certain cyclic subgroups of S n generated by (1 , 2 , ... , k). The analysis of these cyclic subgroups when k is prime reveals an unexpected connection to perfect matchings on a hypercube with certain vertices identified. [ABSTRACT FROM AUTHOR]

We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric functions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. Using this result, we give necessary and sufficient conditions for a dual immaculate quasisymmetric function to be symmetric. Moreover, we show that the product of a Schur function and a dual immaculate quasisymmetric function expands positively in the Young quasisymmetric Schur basis. We also discuss the decomposition of the Young noncommutative Schur functions into the immaculate functions. Finally, we provide a Remmel–Whitney-style rule to generate the coefficients of the decomposition of the dual immaculates into the Young quasisymmetric Schurs algorithmically and an analogous rule for the decomposition of the dual bases. [ABSTRACT FROM AUTHOR]

We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. We examine the combinatorics of the quasisymmetric hook Schur functions, providing a relationship to Gessel's fundamental quasisymmetric functions and an analogue of the Robinson-Schensted-Knuth algorithm. We also prove that the multiplication of quasisymmetric hook Schur functions with hook Schur functions behaves the same as the multiplication of quasisymmetric Schur functions with Schur functions. [ABSTRACT FROM AUTHOR]

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan–Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We point out that, in a certain sense, this formula cannot be simplified. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as a multivariate generating series of nondecreasing functions on the graph. We describe the kernel of this linear map by using a simple combinatorial operation that we call cyclic inclusion-exclusion. Our result also holds for the natural noncommutative analogue and for the commutative and noncommutative restrictions to bipartite graphs. An application to the theory of Kerov character polynomials is given. [ABSTRACT FROM AUTHOR]

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions, called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions, called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as quasisymmetic Schur functions are generated through fillings of composition diagrams. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships. [ABSTRACT FROM AUTHOR]

Abstract: We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall–Littlewood polynomials and Hivertʼs quasisymmetric Hall–Littlewood polynomials . More specifically, we provide: [1.] the G-expansions of the Hall–Littlewood polynomials , the monomial quasisymmetric polynomials , the quasisymmetric Schur polynomials , and the peak quasisymmetric functions ; [2.] an expansion of in terms of the ʼs. The F-expansion of is facilitated by introducing starred tableaux. [Copyright &y& Elsevier]

Let sμ denote a Schur symmetric function and Qα a fundamental quasisymmetric function. Explicit combinatorial formulas are developed for the fundamental quasisymmetric expansions of the plethysms sμ[sν] and sμ[Qα], as well as for related plethysms defined by inequality conditions. The key tools for obtaining these expansions are new standardization and reading word constructions for matrices. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall–Littlewood theory. [ABSTRACT FROM AUTHOR]

Abstract: Every symmetric function can be written uniquely as a linear combination of Schur functions, say , and also as a linear combination of fundamental quasisymmetric functions, say . For many choices of arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients and wishes to compute the Schur coefficients . This paper gives a general combinatorial formula expressing each as a linear combination of the ’s, where each coefficient in this linear combination is , , or 0. This formula arises by suitably modifying Eğecioğlu and Remmel’s combinatorial interpretation of the inverse Kostka matrix involving special rim-hook tableaux. [ABSTRACT FROM AUTHOR]

The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach, we introduce colored analogs of P-partitions and enriched P-partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile’s theory of combinatorial Hopf algebras and its colored analog. [ABSTRACT FROM AUTHOR]

Abstract: We give a combinatorial formula for the inverses of the alternating sums of free quasi-symmetric functions of the form where runs over compositions with parts in a prescribed set . This proves in particular three special cases (no restriction, even parts, and all parts equal to 2) which were conjectured by B.C.V. Ung in [B.C.V. Ung, Combinatorial identities for series of quasi-symmetric functions, in: Proc. FPSAC’08, Toronto, 2008]. [Copyright &y& Elsevier]

Abstract: A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: [ ] defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients; [ ] is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid; [ ] is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight; [ ] behaves simply under matroid duality; [ ] has a simple expansion in terms of -partition enumerators; [ ] is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis. [Copyright &y& Elsevier]