198 research outputs found
Geometric realizations of Tamari interval lattices via cubic coordinates
We introduce cubic coordinates, which are integer words encoding intervals in
the Tamari lattices. Cubic coordinates are in bijection with interval-posets,
themselves known to be in bijection with Tamari intervals. We show that in each
degree the set of cubic coordinates forms a lattice, isomorphic to the lattice
of Tamari intervals. Geometric realizations are naturally obtained by placing
cubic coordinates in space, highlighting some of their properties. We consider
the cellular structure of these realizations. Finally, we show that the poset
of cubic coordinates is shellable
The complexity of counting poset and permutation patterns
We introduce a notion of pattern occurrence that generalizes both classical
permutation patterns as well as poset containment. Many questions about pattern
statistics and avoidance generalize naturally to this setting, and we focus on
functional complexity problems -- particularly those that arise by constraining
the order dimensions of the pattern and text posets. We show that counting the
number of induced, injective occurrences among dimension 2 posets is #P-hard;
enumerating the linear extensions that occur in realizers of dimension 2 posets
can be done in polynomial time, while for unconstrained dimension it is
GI-complete; counting not necessarily induced, injective occurrences among
dimension 2 posets is #P-hard; counting injective or not necessarily injective
occurrences of an arbitrary pattern in a dimension 1 text is #P-hard, although
it is in FP if the pattern poset is constrained to have bounded intrinsic
width; and counting injective occurrences of a dimension 1 pattern in an
arbitrary text is #P-hard, while it is in FP for bounded dimension texts. This
framework easily leads to a number of open questions, chief among which are (1)
is it #P-hard to count the number of occurrences of a dimension 2 pattern in a
dimension 1 text, and (2) is it #P-hard to count the number of texts which
avoid a given pattern?Comment: 15 page
f-vectors of Simplicial Posets that are Balls
Results of R. Stanley and M. Masuda completely characterize the h-vectors of
simplicial posets whose order complexes are spheres. In this paper we examine
the corresponding question in the case where the order complex is a ball. Using
the face rings of these posets, we develop a series of new conditions on their
h-vectors. We also present new methods for constructing poset balls with
specific h-vectors. These results allow us to give a complete characterization
of the h-vectors of simplicial poset balls up through dimension six.Comment: 25 page
On the Topology of the Cambrian Semilattices
For an arbitrary Coxeter group , David Speyer and Nathan Reading defined
Cambrian semilattices as semilattice quotients of the weak order
on induced by certain semilattice homomorphisms. In this article, we define
an edge-labeling using the realization of Cambrian semilattices in terms of
-sortable elements, and show that this is an EL-labeling for every
closed interval of . In addition, we use our labeling to show that
every finite open interval in a Cambrian semilattice is either contractible or
spherical, and we characterize the spherical intervals, generalizing a result
by Nathan Reading.Comment: 20 pages, 5 figure
Combinatorics of the Permutahedra, Associahedra, and Friends
I present an overview of the research I have conducted for the past ten years
in algebraic, bijective, enumerative, and geometric combinatorics. The two main
objects I have studied are the permutahedron and the associahedron as well as
the two partial orders they are related to: the weak order on permutations and
the Tamari lattice. This document contains a general introduction (Chapters 1
and 2) on those objects which requires very little previous knowledge and
should be accessible to non-specialist such as master students. Chapters 3 to 8
present the research I have conducted and its general context. You will find:
* a presentation of the current knowledge on Tamari interval and a precise
description of the family of Tamari interval-posets which I have introduced
along with the rise-contact involution to prove the symmetry of the rises and
the contacts in Tamari intervals;
* my most recent results concerning q, t-enumeration of Catalan objects and
Tamari intervals in relation with triangular partitions;
* the descriptions of the integer poset lattice and integer poset Hopf
algebra and their relations to well known structures in algebraic
combinatorics;
* the construction of the permutree lattice, the permutree Hopf algebra and
permutreehedron;
* the construction of the s-weak order and s-permutahedron along with the
s-Tamari lattice and s-associahedron.
Chapter 9 is dedicated to the experimental method in combinatorics research
especially related to the SageMath software. Chapter 10 describes the outreach
efforts I have participated in and some of my approach towards mathematical
knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche
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