662 research outputs found
The -rational -Catalan polynomials for and their -symmetry
We introduce a new statistic, skip, on rational -Dyck paths and define
a marked rank word for each path when is not a multiple of 3. If a triple
of valid statistics (area,skip,dinv) are given, we have an algorithm to
construct the marked rank word corresponding to the triple. By considering all
valid triples we give an explicit formula for the -rational
-Catalan polynomials when . Then there is a natural bijection on the
triples of statistics (area,skips,dinv) which exchanges the statistics area and
dinv while fixing the skip. Thus we prove the -symmetry of
-rational -Catalan polynomials for .Comment: 11 pages, 4 figure
Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers
We study the statistics area, bounce and dinv on the set of parallelogram
polyominoes having a rectangular m times n bounding box. We show that the
bi-statistics (area, bounce) and (area, dinv) give rise to the same
q,t-analogue of Narayana numbers which was introduced by two of the authors in
[arXiv:1208.0024]. We prove the main conjectures of that paper: the
q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is
accomplished by providing a symmetric functions interpretation of the
q,t-Narayana polynomials which relates them to the famous diagonal harmonics
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Affine permutations and rational slope parking functions
We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund’s bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincaré polynomials of certain affine Springer fibers and describe a connection to the theory of finite-dimensional representations of DAHA and non-symmetric Macdonald polynomials
Type C parking functions and a zeta map
We introduce type C parking functions, encoded as vertically labelled lattice
paths and endowed with a statistic dinv'. We define a bijection from type C
parking functions to regions of the Shi arrangement of type C, encoded as
diagonally labelled ballot paths and endowed with a natural statistic area'.
This bijection is a natural analogue of the zeta map of Haglund and Loehr and
maps dinv' to area'. We give three different descriptions of it.Comment: 12 page
Rational Parking Functions and LLT Polynomials
We prove that the combinatorial side of the "Rational Shuffle Conjecture"
provides a Schur-positive symmetric polynomial. Furthermore, we prove that the
contribution of a given rational Dyck path can be computed as a certain skew
LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel
and Ulyanov. The corresponding skew diagram is described explicitly in terms of
a certain (m,n)-core.Comment: 14 pages, 8 figure
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