291 research outputs found
Cyclic Sieving of Matchings
The cyclic sieving phenomenon (CSP) was introduced by Reiner, Stanton, and
White to study combinatorial structures with actions of cyclic groups. The
crucial step is to find a polynomial, for example a q-analog, that satisfies
the CSP conditions for an action. This polynomial will give us a lot of
information about the symmetry and structure of the set under the action. In
this paper, we study the cyclic sieving phenomenon of the cyclic group
acting on , which is the set of matchings of points on a circle
with crossings. The noncrossing matchings () was recently studied as a
Catalan object. In this paper, we study more general cases, the matchings with
more number of crossings. We prove that there exists -analog polynomials
such that exhibits the cyclic sieving
phenomenon for . In the proof, we also introduce an efficient
representation of the elements in , which helps us to understand the
symmetrical structure of the set
Pattern avoidance in matchings and partitions
Extending the notion of pattern avoidance in permutations, we study matchings
and set partitions whose arc diagram representation avoids a given
configuration of three arcs. These configurations, which generalize 3-crossings
and 3-nestings, have an interpretation, in the case of matchings, in terms of
patterns in full rook placements on Ferrers boards.
We enumerate 312-avoiding matchings and partitions, obtaining algebraic
generating functions, in contrast with the known D-finite generating functions
for the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a
more direct proof of a formula of B\'ona for the number of 1342-avoiding
permutations. Additionally, we give a bijection proving the
shape-Wilf-equivalence of the patterns 321 and 213 which greatly simplifies
existing proofs by Backelin--West--Xin and Jel\'{\i}nek, and provides an
extension of work of Gouyou-Beauchamps for matchings with fixed points.
Finally, we classify pairs of patterns of length 3 according to
shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair
in most of the resulting equivalence classes.Comment: 34 pages, 12 Figures, 5 Table
Crossings and Nestings of Matchings and Partitions
We present results on the enumeration of crossings and nestings for matchings
and set partitions. Using a bijection between partitions and vacillating
tableaux, we show that if we fix the sets of minimal block elements and maximal
block elements, the crossing number and the nesting number of partitions have a
symmetric joint distribution. It follows that the crossing numbers and the
nesting numbers are distributed symmetrically over all partitions of , as
well as over all matchings on . As a corollary, the number of
-noncrossing partitions is equal to the number of -nonnesting partitions.
The same is also true for matchings. An application is given to the enumeration
of matchings with no -crossing (or with no -nesting).Comment: Revision: page 8, revised Remark (3) page 11, revised Proof of
Theorem 1. Page 20, revised Remark. page 23, added reference
A Bijection for Crossings and Nestings
For a subclass of matchings, set partitions, and permutations, we describe a
direct bijection involving only arc annotated diagrams that not only
interchanges maximum nesting and crossing numbers, but also all refinements of
crossing and nesting numbers. Furthermore, we show that the bijection cannot be
applied to a similar class of coloured arc annotated diagrams
Fully packed loop configurations: polynomiality and nested arches
This article proves a conjecture by Zuber about the enumeration of fully
packed loops (FPLs). The conjecture states that the number of FPLs whose link
pattern consists of two noncrossing matchings which are separated by nested
arches is a polynomial function in of certain degree and with certain
leading coefficient. Contrary to the approach of Caselli, Krattenthaler, Lass
and Nadeau (who proved a partial result) we make use of the theory of wheel
polynomials developed by Di Francesco, Fonseca and Zinn-Justin. We present a
new basis for the vector space of wheel polynomials and a polynomiality theorem
in a more general setting. This allows us to finish the proof of Zubers
conjecture.Comment: replaced with revised versio
Springer representations on the Khovanov Springer varieties
Springer varieties are studied because their cohomology carries a natural
action of the symmetric group and their top-dimensional cohomology is
irreducible. In his work on tangle invariants, Khovanov constructed a family of
Springer varieties as subvarieties of the product of spheres .
We show that if is embedded antipodally in then the natural
-action on induces an -representation on the image of
. This representation is the Springer representation. Our
construction admits an elementary (and geometrically natural) combinatorial
description, which we use to prove that the Springer representation on
is irreducible in each degree. We explicitly identify the
Kazhdan-Lusztig basis for the irreducible representation of corresponding
to the partition
A topological construction for all two-row Springer varieties
Springer varieties appear in both geometric representation theory and knot
theory. Motivated by knot theory and categorification Khovanov provides a
topological construction of Springer varieties. We extend
Khovanov's construction to all two-row Springer varieties. Using the
combinatorial and diagrammatic properties of this construction we provide a
particularly useful homology basis and construct the Springer representation
using this basis. We also provide a skein-theoretic formulation of the
representation in this case.Comment: 27 pages, many figure
Determinant Formulas Relating to Tableaux of Bounded Height
Chen et al. recently established bijections for -noncrossing/
nonnesting matchings, oscillating tableaux of bounded height , and
oscillating lattice walks in the -dimensional Weyl chamber. Stanley asked
what is the total number of such tableaux of length and of any shape. We
find a determinant formula for the exponential generating function. The same
idea applies to prove Gessel's remarkable determinant formula for permutations
with bounded length of increasing subsequences. We also give short algebraic
derivations for some results of the reflection principle.Comment: 15 pages, 4 figure
Counting partitions of a fixed genus
We show that, for any fixed genus , the ordinary generating function for
the genus partitions of an -element set into blocks is algebraic.
The proof involves showing that each such partition may be reduced in a unique
way to a primitive partition and that the number of primitive partitions of a
given genus is finite. We illustrate our method by finding the generating
function for genus partitions, after identifying all genus primitive
partitions, using a computer-assisted search
Polynomials, meanders, and paths in the lattice of noncrossing partitions
For every polynomial f of degree n with no double roots, there is an
associated family C(f) of harmonic algebraic curves, fibred over the circle,
with at most n-1 singular fibres. We study the combinatorial topology of C(f)
in the generic case when there are exactly n-1 singular fibres. In this case,
the topology of C(f) is determined by the data of an n-tuple of noncrossing
matchings on the set {0,1,...,2n-1} with certain extra properties. We prove
that there are 2(2n)^{n-2} such n-tuples, and that all of them arise from the
topology of C(f) for some polynomial f.Comment: 24 pages, 7 figures. To appear, Transactions of the A.M.S. Revised
based on referee report; final section adde
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