65 research outputs found
The number of directed k-convex polyominoes
We present a new method to obtain the generating functions for directed
convex polyominoes according to several different statistics including: width,
height, size of last column/row and number of corners. This method can be used
to study different families of directed convex polyominoes: symmetric
polyominoes, parallelogram polyominoes. In this paper, we apply our method to
determine the generating function for directed k-convex polyominoes. We show it
is a rational function and we study its asymptotic behavior
The excedances and descents of bi-increasing permutations
Starting from some considerations we make about the relations between certain
difference statistics and the classical permutation statistics we study
permutations whose inversion number and excedance difference coincide. It turns
out that these (so-called bi-increasing) permutations are just the 321-avoiding
ones. The paper investigates their excedance and descent structure. In
particular, we find some nice combinatorial interpretations for the
distribution coefficients of the number of excedances and descents,
respectively, and their difference analogues over the bi-increasing
permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This
yields a connection between restricted permutations, parallelogram polyominoes,
and lattice paths that reveals the relations between several well-known
bijections given for these objects (e.g. by Delest-Viennot,
Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an
application, we enumerate skew diagrams according to their rank and give a
simple combinatorial proof for a result concerning the symmetry of the joint
distribution of the number of excedances and inversions, respectively, over the
symmetric group.Comment: 36 page
Solving multivariate functional equations
This paper presents a new method to solve functional equations of
multivariate generating functions, such as
giving a
formula for in terms of a sum over finite sequences. We use this
method to show how one would calculate the coefficients of the generating
function for parallelogram polyominoes, which is impractical using other
methods. We also apply this method to answer a question from fully commutative
affine permutations.Comment: 11 pages, 1 figure. v3: Main theorems and writing style revised for
greater clarity. Updated to final version, to appear in Discrete Mathematic
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
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
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
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