80 research outputs found
Ballot tilings and increasing trees
We study enumerations of Dyck and ballot tilings, which are tilings of a
region determined by two Dyck or ballot paths. We give bijective proofs to two
formulae of enumerations of Dyck tilings through Hermite histories. We show
that one of the formulae is equal to a certain Kazhdan--Lusztig polynomial. For
a ballot tiling, we establish formulae which are analogues of formulae for Dyck
tilings. Especially, the generating functions have factorized expressions. The
key tool is a planted plane tree and its increasing labellings. We also
introduce a generalized perfect matching which is bijective to an Hermite
history for a ballot tiling. By combining these objects, we obtain various
expressions of a generating function of ballot tilings with a fixed lower path.Comment: 53 page
On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
We prove that the Cayley graph and the coset geometry of the von Dyck group
are linked by a vertex-to-edge duality.Comment: Accepted for publication on Mathematica Slovaca (23-10-2013); 12
pages, 7 figures; completely rewritten replacement of "On the free Burnside
group as a factor of von Dyck group: a geometric perspective
Symmetric Dyck tilings, ballot tableaux and tree-like tableaux of shifted shapes
Symmetric Dyck tilings and ballot tilings are certain tilings in the region
surrounded by two ballot paths. We study the relations of combinatorial objects
which are bijective to symmetric Dyck tilings such as labeled trees, Hermite
histories, and perfect matchings. We also introduce two operations on labeled
trees for symmetric Dyck tilings: symmetric Dyck tiling strip (symDTS) and
symmetric Dyck tiling ribbon (symDTR). We give two definitions of Hermite
histories for symmetric Dyck tilings, and show that they are equivalent by use
of the correspondence between symDTS operation and an Hermite history. Since
ballot tilings form a subset in the set of symmetric Dyck tilings, we construct
an inclusive map from labeled trees for ballot tilings to labeled trees for
symmetric Dyck tilings. By this inclusive map, the results for symmetric Dyck
tilings can be applied to those of ballot tilings. We introduce and study the
notions of ballot tableaux and tree-like tableaux of shifted shapes, which are
generalizations of Dyck tableaux and tree-like tableaux, respectively. The
correspondence between ballot tableaux and tree-like tableaux of shifted shapes
is given by using the symDTR operation and the structure of labeled trees for
symmetric Dyck tilings.Comment: 60 page
Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation
with reflecting boundary conditions which is relevant to the Temperley--Lieb
model of loops on a strip. By use of integral formulae we prove conjectures
relating it to the weighted enumeration of Cyclically Symmetric Transpose
Complement Plane Partitions and related combinatorial objects
Enumeration of colored Dyck paths via partial Bell polynomials
We consider a class of lattice paths with certain restrictions on their
ascents and down steps and use them as building blocks to construct various
families of Dyck paths. We let every building block take on colors
and count all of the resulting colored Dyck paths of a given semilength. Our
approach is to prove a recurrence relation of convolution type, which yields a
representation in terms of partial Bell polynomials that simplifies the
handling of different colorings. This allows us to recover multiple known
formulas for Dyck paths and related lattice paths in an unified manner.Comment: 10 pages. Submitted for publicatio
Enumerations of humps and peaks in -paths and -Dyck paths via bijective proofs
Recently Mansour and Shattuck studied -paths and gave formulas that
relate the total number of humps (peaks) in all -paths to the number of
super -paths. These results generalize earlier results of Regev on Dyck
paths and Motzkin paths. Their proofs are based on generating functions and
they asked for bijective proofs for their results. In this paper we first give
bijective proofs of Mansour and Shattuck's results, then we extend our study to
-Dyck paths. We give a bijection that relates the total number of peaks
in all -Dyck paths to certain free -paths when and are
coprime. From this bijection we get the number of -Dyck paths with
exactly peaks, which is a generalization of the well-known result that the
number Dyck paths of order with exactly peaks is the Narayana number
.Comment: 11 page
Deforming the Fredkin spin chain away from its frustration-free point
The Fredkin model describes a spin-half chain segment subject to three-body,
correlated-exchange interactions and twisted boundary conditions. The model is
frustration-free, and its ground state wave function is known exactly. Its
low-energy physics is that of a strong xy ferromagnet with gapless excitations
and an unusually large dynamical exponent. We study a generalized spin chain
model that includes the Fredkin model as a special tuning point and otherwise
interpolates between the conventional ferromagnetic and antiferromagnetic
quantum Heisenberg models. We solve for the low-lying states, using exact
diagonalization and density-matrix renormalization group calculations, in order
to track the properties of the system as it is tuned away from the Fredkin
point; we also present exact analytical results that hold right at the Fredkin
point. We identify a zero-temperature phase diagram with multiple transitions
and unexpected ordered phases. The Fredkin ground state turns out to be
particularly brittle, unstable to even infinitesimal antiferromagnetic
frustration. We remark on the existence of an "anti-Fredkin" point at which all
the contributing spin configurations have a spin structure exactly opposite to
those in the Fredkin ground state.Comment: 11 pages, 14 figures; new simulations and analysis of the gap
scaling; matches the published versio
Dyck tilings of type
We introduce and study cover-inclusive and cover-exclusive Dyck tilings of
type . It is shown that the generating functions of Dyck tilings of type
are expressed in terms of the generating function of ballot tilings of type
. We introduce link patterns of type and plane trees for a ballot path,
and construct a map from trees to . This map gives the
generating function of cover-inclusive Dyck tilings of type associated to
the ballot path.Comment: 14 page
Bijections on -Shi and -Catalan Arrangements
Associated with the -Shi arrangement and -Catalan arrangement in
, we introduce a cubic matrix for each region to establish two
bijections in a uniform way. Firstly, the positions of minimal positive entries
in column slices of the cubic matrix will give a bijection from regions of the
-Shi arrangement to -rooted labeled -trees. Secondly, the numbers of
positive entries in column slices of the cubic matrix will give a bijection
from regions of the -Catalan arrangement to pairings of permutation and
-Dyck path. Moreover, the numbers of positive entries in row slices of the
cubic matrix will recover the Pak-Stanley labeling, a celebrated bijection from
regions of the -Shi arrangement to -parking functions.Comment: 19 pages, 2 figure
Fully Packed Loops in a triangle: matchings, paths and puzzles
Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the
study of ordinary Fully Packed Loop configurations (FPLs) on the square grid
where they were used to show that the number of FPLs with a given link pattern
that has m nested arches is a polynomial function in m. It soon turned out that
TFPLs possess a number of other nice properties. For instance, they can be seen
as a generalized model of Littlewood-Richardson coefficients. We start our
article by introducing oriented versions of TFPLs; their main advantage in
comparison with ordinary TFPLs is that they involve only local constraints.
Three main contributions are provided. Firstly, we show that the number of
ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs
and thus it suffices to consider the latter. Secondly, we decompose oriented
TFPLs into two matchings and use a classical bijection to obtain two families
of nonintersecting lattice paths (path tangles). This point of view turns out
to be extremely useful for giving easy proofs of previously known conditions on
the boundary of TFPLs necessary for them to exist. One example is the
inequality d(u)+d(v)<=d(w) where u,v,w are 01-words that encode the boundary
conditions of ordinary TFPLs and d(u) is the number of cells in the Ferrers
diagram associated with u. In the third part we consider TFPLs with d(w)-
d(u)-d(v)=0,1; in the first case their numbers are given by
Littlewood-Richardson coefficients, but also in the second case we provide
formulas that are in terms of Littlewood-Richardson coefficients. The proofs of
these formulas are of a purely combinatorial nature.Comment: 40 pages, 31 figure
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