44 research outputs found
Yay for Determinants!
In this {\it case study}, we hope to show why Sheldon Axler was not just
wrong, but {\em wrong}, when he urged, in 1995: ``Down with Determinants!''. We
first recall how determinants are useful in enumerative combinatorics, and then
illustrate three versatile tools (Dodgson's condensation, the holonomic ansatz
and constant term evaluations) to operate in tandem to prove a certain
intriguing determinantal formula conjectured by the first author.Comment: 13 page
Hook formulas for skew shapes
International audienceThe celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations
Maximal 0-1 fillings of moon polyominoes with restricted chain-lengths and rc-graphs
We show that maximal 0-1-fillings of moon polynomials with restricted chain
lengths can be identified with certain rc-graphs, also known as pipe dreams. In
particular, this exhibits a connection between maximal 0-1-fillings of Ferrers
shapes and Schubert polynomials. Moreover, it entails a bijective proof showing
that the number of maximal fillings of a stack polyomino S with no north-east
chains longer than k depends only on k and the multiset of column heights of S.
Our main contribution is a slightly stronger theorem, which in turn leads us to
conjecture that the poset of rc-graphs with covering relation given by
generalised chute moves is in fact a lattice.Comment: 22 pages, v2: references added, v3: included proof for bijection for
stack polyominoes, v4: include conjecture and improve presentatio
Multi-cluster complexes
We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types and to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex
On Convex Geometric Graphs with no Pairwise Disjoint Edges
A well-known result of Kupitz from 1982 asserts that the maximal number of
edges in a convex geometric graph (CGG) on vertices that does not contain
pairwise disjoint edges is (provided ). For and
, the extremal examples are completely characterized. For all other
values of , the structure of the extremal examples is far from known: their
total number is unknown, and only a few classes of examples were presented,
that are almost symmetric, consisting roughly of the "longest possible"
edges of , the complete CGG of order .
In order to understand further the structure of the extremal examples, we
present a class of extremal examples that lie at the other end of the spectrum.
Namely, we break the symmetry by requiring that, in addition, the graph admit
an independent set that consists of consecutive vertices on the boundary of
the convex hull. We show that such graphs exist as long as and
that this value of is optimal.
We generalize our discussion to the following question: what is the maximal
possible number of edges in a CGG on vertices that does not
contain pairwise disjoint edges, and, in addition, admits an independent
set that consists of consecutive vertices on the boundary of the convex
hull? We provide a complete answer to this question, determining for
all relevant values of and .Comment: 17 pages, 9 figure
A Hopf algebra of subword complexes (Extended abstract)
International audienceWe introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras
Maximal increasing sequences in fillings of almost-moon polyominoes
It was proved by Rubey that the number of fillings with zeros and ones of a
given moon polyomino that do not contain a northeast chain of size depends
only on the set of columns of the polyomino, but not the shape of the
polyomino. Rubey's proof is an adaption of jeu de taquin and promotion for
arbitrary fillings of moon polyominoes. In this paper we present a bijective
proof for this result by considering fillings of almost-moon polyominoes, which
are moon polyominoes after removing one of the rows. Explicitly, we construct a
bijection which preserves the size of the largest northeast chains of the
fillings when two adjacent rows of the polyomino are exchanged. This bijection
also preserves the column sum of the fillings. We also present a bijection that
preserves the size of the largest northeast chains, the row sum and the column
sum if every row of the fillings has at most one 1.Comment: 18 page