367 research outputs found

    Eulerian quasisymmetric functions and cyclic sieving

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    It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group SnS_n generated by the nn-cycle (1,2,...,n)(1,2,...,n) on the set of permutations of fixed cycle type and fixed number of excedances provides an instance of the cyclic sieving phenonmenon of Reiner, Stanton and White. The main tool is a class of symmetric functions recently introduced in work of two of the authors.Comment: 30 page

    Refined Catalan and Narayana cyclic sieving

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    We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.Comment: Updated version, minor change

    Cyclic sieving, skew Macdonald polynomials and Schur positivity

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    When λ\lambda is a partition, the specialized non-symmetric Macdonald polynomial Eλ(x;q;0)E_{\lambda}(x;q;0) is symmetric and related to a modified Hall--Littlewood polynomial. We show that whenever all parts of the integer partition λ\lambda is a multiple of nn, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under a cyclic shift of the columns. The corresponding CSP polynomial is given by Eλ(x;q;0)E_{\lambda}(x;q;0). In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B.~Rhoades. We also introduce a skew version of Eλ(x;q;0)E_{\lambda}(x;q;0). We show that these are symmetric and Schur-positive via a variant of the Robinson--Schenstedt--Knuth correspondence and we also describe crystal raising- and lowering operators for the underlying fillings. Moreover, we show that the skew specialized non-symmetric Macdonald polynomials are in some cases vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials

    The cyclic sieving phenomenon: a survey

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    The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many instances of the cyclic sieving phenomenon have now been found. Furthermore, the proofs that this phenomenon hold often involve interesting and sometimes deep results from representation theory. We will survey the current literature on cyclic sieving, providing the necessary background about representations, Coxeter groups, and other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes suggested by colleagues and the referee. To appear in the London Mathematical Society Lecture Note Series. The third version has a few smaller change

    Equidistribution and Sign-Balance on 321-Avoiding Permutations

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    Let TnT_n be the set of 321-avoiding permutations of order nn. Two properties of TnT_n are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over TnT_n, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for Dyck paths. (2) The sign-and-last-descent enumerators for T2nT_{2n} and T2n+1T_{2n+1} are essentially equal to the last-descent enumerator for TnT_n. The proofs use a recursion formula for an appropriate multivariate generating function.Comment: 17 pages; to appear in S\'em. Lothar. Combi
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