367 research outputs found
Eulerian quasisymmetric functions and cyclic sieving
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 generated by the -cycle 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
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
When is a partition, the specialized non-symmetric Macdonald
polynomial is symmetric and related to a modified
Hall--Littlewood polynomial. We show that whenever all parts of the integer
partition is a multiple of , 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 . 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 . 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
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
Let be the set of 321-avoiding permutations of order . Two
properties of are proved: (1) The {\em last descent} and {\em last index
minus one} statistics are equidistributed over , 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
and are essentially equal to the last-descent enumerator
for . 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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