21 research outputs found
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
Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial
We classify recurrent configurations of the sandpile model on the complete
bipartite graph K_{m,n} in which one designated vertex is a sink. We present a
bijection from these recurrent configurations to decorated parallelogram
polyominoes whose bounding box is a m*n rectangle. Several special types of
recurrent configurations and their properties via this bijection are examined.
For example, recurrent configurations whose sum of heights is minimal are shown
to correspond to polyominoes of least area. Two other classes of recurrent
configurations are shown to be related to bicomposition matrices, a matrix
analogue of set partitions, and (2+2)-free partially ordered sets.
A canonical toppling process for recurrent configurations gives rise to a
path within the associated parallelogram polyominoes. This path bounces off the
external edges of the polyomino, and is reminiscent of Haglund's well-known
bounce statistic for Dyck paths. We define a collection of polynomials that we
call q,t-Narayana polynomials, defined to be the generating function of the
bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to
the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In
doing so, we have extended a bistatistic of Egge, Haglund, Kremer and
Killpatrick (2003) to the set of parallelogram polyominoes. This is one answer
to their question concerning extensions to other combinatorial objects.
We conjecture the q,t-Narayana polynomials to be symmetric and prove this
conjecture for numerous special cases. We also show a relationship between
Haglund's (area,hagbounce) statistic on Dyck paths, and our bistatistic
(area,parabounce) on a sub-collection of those parallelogram polyominoes living
in a (n+1)*n rectangle
Combinatorics of Labelled Parallelogram polyominoes
We obtain explicit formulas for the enumeration of labelled parallelogram
polyominoes. These are the polyominoes that are bounded, above and below, by
north-east lattice paths going from the origin to a point (k,n). The numbers
from 1 and n (the labels) are bijectively attached to the north steps of
the above-bounding path, with the condition that they appear in increasing
values along consecutive north steps. We calculate the Frobenius characteristic
of the action of the symmetric group S_n on these labels. All these enumeration
results are refined to take into account the area of these polyominoes. We make
a connection between our enumeration results and the theory of operators for
which the intergral Macdonald polynomials are joint eigenfunctions. We also
explain how these same polyominoes can be used to explicitly construct a linear
basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure
Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma
We introduce two operators on stable configurations of the sandpile model
that provide an algorithmic bijection between recurrent and parking
configurations. This bijection preserves their equivalence classes with respect
to the sandpile group. The study of these operators in the special case of the
complete bipartite graph naturally leads to a generalization of the
well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic
bi-infinite paths in the plane having slightly different slopes. We achieve our
results by interpreting the action of these operators as an action on a point
in the grid which is pointed to by one of these pairs of paths.
Our Cyclic lemma allows us to enumerate several classes of polyominoes, and
therefore builds on the work of Irving and Rattan (2009), Chapman et al.
(2009), and Bonin et al. (2003).Comment: 28 page
A refinement of the Shuffle Conjecture with cars of two sizes and
The original Shuffle Conjecture of Haglund et al. has a symmetric function
side and a combinatorial side. The symmetric function side may be simply
expressed as where \nabla is the Macdonald polynomial
eigen-operator of Bergeron and Garsia and is the homogeneous basis
indexed by partitions of n. The combinatorial
side q,t-enumerates a family of Parking Functions whose reading word is a
shuffle of k successive segments of 1,2,3,...,n of respective lengths
. It can be shown that for t=1/q the symmetric function
side reduces to a product of q-binomial coefficients and powers of q. This
reduction suggests a surprising combinatorial refinement of the general Shuffle
Conjecture. Here we prove this refinement for k=2 and t=1/q. The resulting
formula gives a q-analogue of the well studied Narayana numbers.Comment: 17 pages, 11 figure