52 research outputs found
An Introduction to Combinatorics via Cayley\u27s Theorem
In this paper, we explore some of the methods that are often used to solve combinatorial problems by proving Cayley’s theorem on trees in multiple ways. The intended audience of this paper is undergraduate and graduate mathematics students with little to no experience in combinatorics. This paper could also be used as a supplementary text for an undergraduate combinatorics course
Bijections for Baxter Families and Related Objects
The Baxter number can be written as . These
numbers have first appeared in the enumeration of so-called Baxter
permutations; is the number of Baxter permutations of size , and
is the number of Baxter permutations with descents and
rises. With a series of bijections we identify several families of
combinatorial objects counted by the numbers . Apart from Baxter
permutations, these include plane bipolar orientations with vertices and
faces, 2-orientations of planar quadrangulations with white and
black vertices, certain pairs of binary trees with left and
right leaves, and a family of triples of non-intersecting lattice paths. This
last family allows us to determine the value of as an
application of the lemma of Gessel and Viennot. The approach also allows us to
count certain other subfamilies, e.g., alternating Baxter permutations, objects
with symmetries and, via a bijection with a class of plan bipolar orientations
also Schnyder woods of triangulations, which are known to be in bijection with
3-orientations.Comment: 31 pages, 22 figures, submitted to JCT
Path Tableaux and the Combinatorics of the Immanant Function
Immanants are a generalization of the well-studied determinant and permanent. Although the combinatorial interpretations for the determinant and permanent have been studied in excess, there remain few combinatorial interpretations for the immanant.
The main objective of this thesis is to consider the immanant, and its possible combinatorial interpretations, in terms of recursive structures on the character. This thesis presents a comprehensive view of previous interpretations of immanants. Furthermore, it discusses algebraic techniques that may be used to investigate further into the combinatorial aspects of the immanant.
We consider the Temperley-Lieb algebra and the class of immanants over the elements of this algebra. Combinatorial tools including the Temperley-Lieb algebra and Kauffman diagrams will be used in a number of interpretations. In particular, we extend some results for the permanent and determinant based on the -weighted planar network construction, where is a convenient ring, by Clearman, Shelton, and Skandera. This thesis also presents some cases in which this construction cannot be extended. Finally, we present some extensions to combinatorial interpretations on certain classes of tableaux, as well as certain classes of matrices
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
Catalan's intervals and realizers of triangulations
The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable
orders defined on the set of Catalan objects of a given size. These lattices
are ordered by inclusion: the Stanley lattice is an extension of the Tamari
lattice which is an extension of the Kreweras lattice. The Stanley order can be
defined on the set of Dyck paths of size as the relation of \emph{being
above}. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck
paths. In a former article, the second author defined a bijection
between pairs of non-crossing Dyck paths and the realizers of triangulations
(or Schnyder woods). We give a simpler description of the bijection .
Then, we study the restriction of to Tamari's and Kreweras' intervals.
We prove that induces a bijection between Tamari intervals and minimal
realizers. This gives a bijection between Tamari intervals and triangulations.
We also prove that induces a bijection between Kreweras intervals and
the (unique) realizers of stack triangulations. Thus, induces a
bijection between Kreweras intervals and stack triangulations which are known
to be in bijection with ternary trees.Comment: 22 page
The two periodic Aztec diamond and matrix valued orthogonal polynomials
We analyze domino tilings of the two-periodic Aztec diamond by means of
matrix valued orthogonal polynomials that we obtain from a reformulation of the
Aztec diamond as a non-intersecting path model with periodic transition
matrices. In a more general framework we express the correlation kernel for the
underlying determinantal point process as a double contour integral that
contains the reproducing kernel of matrix valued orthogonal polynomials. We use
the Riemann-Hilbert problem to simplify this formula for the case of the
two-periodic Aztec diamond.
In the large size limit we recover the three phases of the model known as
solid, liquid and gas. We describe fine asymptotics for the gas phase and at
the cusp points of the liquid-gas boundary, thereby complementing and extending
results of Chhita and Johansson.Comment: 80 pages, 20 figures; This is an extended version of the paper that
is accepted for publication in the Journal of the EM
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