8 research outputs found
Computing growth functions of braid monoids and counting vertex-labelled bipartite graphs
We derive a recurrence relation for the number of simple vertex-labelled
bipartite graphs with given degrees of the vertices and use this result to
obtain a new method for computing the growth function of the Artin monoid of
type with respect to the simple elements (permutation braids) as
generators. Instead of matrices of size , we use
matrices of size , where is the number of partitions of
.Comment: reference adde
A divisibility result on combinatorics of generalized braids
For every finite Coxeter group , each positive braids in the
corresponding braid group admits a unique decomposition as a finite sequence of
elements of , the so-called Garside-normal form.The study of the
associated adjacency matrix allows to count the number of
Garside-normal form of a given length.In this paper we prove that the
characteristic polynomial of divides the one of . The
key point is the use of a Hopf algebra based on signed permutations. A similar
result was already known for the type . We observe that this does not hold
for type . The other Coxeter types (, , and ) are also studied.Comment: 28 page
Recommended from our members
Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
Introduction to Vassiliev Knot Invariants
This book is a detailed introduction to the theory of finite type (Vassiliev)
knot invariants, with a stress on its combinatorial aspects. It is intended to
serve both as a textbook for readers with no or little background in this area,
and as a guide to some of the more advanced material. Our aim is to lead the
reader to understanding by means of pictures and calculations, and for this
reason we often prefer to convey the idea of the proof on an instructive
example rather than give a complete argument. While we have made an effort to
make the text reasonably self-contained, an advanced reader is sometimes
referred to the original papers for the technical details of the proofs.
Version 3: some typos and inaccuracies are corrected.Comment: 512 pages, thousands picture
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282