5,449 research outputs found
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
The representation of the symmetric group on m-Tamari intervals
An m-ballot path of size n is a path on the square grid consisting of north
and east unit steps, starting at (0,0), ending at (mn,n), and never going below
the line {x=my}. The set of these paths can be equipped with a lattice
structure, called the m-Tamari lattice and denoted by T_n^{m}, which
generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was
introduced by F. Bergeron in connection with the study of diagonal coinvariant
spaces in three sets of n variables. The representation of the symmetric group
S_n on these spaces is conjectured to be closely related to the natural
representation of S_n on (labelled) intervals of the m-Tamari lattice, which we
study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north
steps of Q are labelled from 1 to n in such a way the labels increase along any
sequence of consecutive north steps. The symmetric group S_n acts on labelled
intervals of T_n^{m} by permutation of the labels. We prove an explicit
formula, conjectured by F. Bergeron and the third author, for the character of
the associated representation of S_n. In particular, the dimension of the
representation, that is, the number of labelled m-Tamari intervals of size n,
is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The
form of these numbers suggests a connection with parking functions, but our
proof is not bijective. The starting point is a recursive description of
m-Tamari intervals. It yields an equation for an associated generating
function, which is a refined version of the Frobenius series of the
representation. This equation involves two additional variables x and y, a
derivative with respect to y and iterated divided differences with respect to
x. The hardest part of the proof consists in solving it, and we develop
original techniques to do so, partly inspired by previous work on polynomial
equations with "catalytic" variables.Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398,
which will not be submitted to any journa
A polytope related to empirical distributions, plane trees, parking functions, and the associahedron
We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0,
whose combinatorial properties are closely connected with empirical
distributions, plane trees, plane partitions, parking functions, and the
associahedron. In particular, we give explicit formulas for the volume of
Pi_n(x) and, when the x_i's are integers, the number of integer points in
Pi_n(x). We give two polyhedral decompositions of Pi_n(x), one related to order
cones of posets and the other to the associahedron.Comment: 41 page
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
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